Energetic electron motion in the geomagnetic field Energetic - - PowerPoint PPT Presentation

Jaejin Lee Jaejin Lee Jaejin Lee Jaejin Lee (Korea Astronomy and Space science Institute) (Korea Astronomy and Space science Institute) (Korea Astronomy and Space science Institute) (Korea Astronomy and Space science Institute)

Energetic electron motion in the geomagnetic field Energetic electron motion in the geomagnetic field Energetic electron motion in the geomagnetic field Energetic electron motion in the geomagnetic field

• Test particle simulation and observations

Test particle simulation and observations Test particle simulation and observations Test particle simulation and observations -

• Laboratory, Space, and Astrophysical Plasma Workshop

Laboratory, Space, and Astrophysical Plasma Workshop Laboratory, Space, and Astrophysical Plasma Workshop Laboratory, Space, and Astrophysical Plasma Workshop

• Feb. 22, 2009
• Feb. 22, 2009
• Feb. 22, 2009
• Feb. 22, 2009

APL

In My Talk, In My Talk, In My Talk, In My Talk, How energetic electrons are accelerated How energetic electrons are accelerated How energetic electrons are accelerated How energetic electrons are accelerated How they are lost (precipitation) from the radiation belt How they are lost (precipitation) from the radiation belt How they are lost (precipitation) from the radiation belt How they are lost (precipitation) from the radiation belt How they interact with waves How they interact with waves How they interact with waves How they interact with waves

• Discovery of Van Allen radiation

belts – Explorer 1, 1958

• Trapped protons & electrons,

spatial distribution (2-7 RE), energy (~MeV)

Radiation Belt Radiation Belt Radiation Belt Radiation Belt

Three Adiabatic Invariant Three Adiabatic Invariant Three Adiabatic Invariant Three Adiabatic Invariant

First Adiabatic Invariant Second Adiabatic Invariant Third Adiabatic Invariant Total magnetic flux Φ enclosed by a drift surface

• ⊥

=

• ∫

=

Charged Particle Motion in Magnetosphere

Gyro, bounce and drift motions

Gyro ~millisecond, bounce ~ 0.1-1 second, drift ~1-10 minutes

To change particle energy, must violate one or more invariants

Sudden changes of field configurations Small but periodic variation of field configurations

It It It It’ ’ ’ ’s still open problem. s still open problem. s still open problem. s still open problem.

Proposed physical processes

Acceleration Acceleration Acceleration Acceleration: large- and small-scale recirculations, heating by Whistler waves, radial diffusion by ULF waves, cusp source, substorm injection, sudden impulse of solar wind pressure and etc. Loss Loss Loss Loss: pitch angle diffusion, Coulomb collision, and Magnetopause shadowing. Transport Transport Transport Transport

• +

× =

• +

=

+

Euler method Euler method Euler method Euler method Runge Runge Runge Runge-

• Kutta method

Kutta method Kutta method Kutta method

Solving Ordinary Derivative Equation Solving Ordinary Derivative Equation Solving Ordinary Derivative Equation Solving Ordinary Derivative Equation

• +

+ = + + = =

+

Loss Process Loss Process Loss Process Loss Process

Pitch Angle Diffusion by Field Line Curvature Pitch Angle Diffusion by Field Line Curvature Pitch Angle Diffusion by Field Line Curvature Pitch Angle Diffusion by Field Line Curvature

Relativistic Electron Dropouts (RED) Relativistic Electron Dropouts (RED) Relativistic Electron Dropouts (RED) Relativistic Electron Dropouts (RED)

LANL data

Electron Dropout Sequence Electron Dropout Sequence Electron Dropout Sequence Electron Dropout Sequence

Space Space Space Space Environment Environment Environment Environment

Date (Jun 2004) Date (Jun 2004) Date (Jun 2004) Date (Jun 2004)

• 1. Fast dropouts for 1 ~ 5 hours
• 1. Fast dropouts for 1 ~ 5 hours
• 1. Fast dropouts for 1 ~ 5 hours
• 1. Fast dropouts for 1 ~ 5 hours
• 2. Observed for both of electrons and protons
• 2. Observed for both of electrons and protons
• 2. Observed for both of electrons and protons
• 2. Observed for both of electrons and protons
• 3. More effective for higher energy charged particles
• 3. More effective for higher energy charged particles
• 3. More effective for higher energy charged particles
• 3. More effective for higher energy charged particles
• 4. Started from dusk and midnight sector and
• 4. Started from dusk and midnight sector and
• 4. Started from dusk and midnight sector and
• 4. Started from dusk and midnight sector and

propagate to noon sector propagate to noon sector propagate to noon sector propagate to noon sector

• 5. Correlation with magnetic field stretching
• 5. Correlation with magnetic field stretching
• 5. Correlation with magnetic field stretching
• 5. Correlation with magnetic field stretching

Relativistic electron dropout Relativistic electron dropout Relativistic electron dropout Relativistic electron dropout characteristics characteristics characteristics characteristics

Electron Loss Electron Loss Electron Loss Electron Loss

Escape from the magnetopause Escape from the magnetopause Escape from the magnetopause Escape from the magnetopause

• Magnetopause can be compressed inside L = 6.6
• De-trapping of particles and drift outward to magnetopause

Loss to the atmosphere Loss to the atmosphere Loss to the atmosphere Loss to the atmosphere

• Pitch angle scattering into the loss cone
• Observation of particle precipitation

First Adiabatic Invariant Violation First Adiabatic Invariant Violation First Adiabatic Invariant Violation First Adiabatic Invariant Violation

• =

= = ρ ρ κ

Pitch angle diffusion by field line curvature Pitch angle diffusion by field line curvature Pitch angle diffusion by field line curvature Pitch angle diffusion by field line curvature

First Adiabatic Invariant Violation First Adiabatic Invariant Violation First Adiabatic Invariant Violation First Adiabatic Invariant Violation

First adiabatic invariant : The magnetic moment of gyrating part First adiabatic invariant : The magnetic moment of gyrating part First adiabatic invariant : The magnetic moment of gyrating part First adiabatic invariant : The magnetic moment of gyrating particle is icle is icle is icle is conserved conserved conserved conserved as long as magnetic field is constant during a gyro period. as long as magnetic field is constant during a gyro period. as long as magnetic field is constant during a gyro period. as long as magnetic field is constant during a gyro period.

Loss cone filling (Particle simulation) Loss cone filling (Particle simulation) Loss cone filling (Particle simulation) Loss cone filling (Particle simulation)

Electrons of 0.0075% are lost by precipitation. dF/dt = (-2 x 0.000075) F / T_b E-folding loss time : ~ 3.1 hr

• Correlation with magnetic field stretching

Correlation with magnetic field stretching Correlation with magnetic field stretching Correlation with magnetic field stretching

Precipitations by curvature explains a lot of things of RED Precipitations by curvature explains a lot of things of RED Precipitations by curvature explains a lot of things of RED Precipitations by curvature explains a lot of things of RED event event event event

• Fast loss process for 1 ~ 5 hours

Fast loss process for 1 ~ 5 hours Fast loss process for 1 ~ 5 hours Fast loss process for 1 ~ 5 hours

• Observed for both of electrons and protons

Observed for both of electrons and protons Observed for both of electrons and protons Observed for both of electrons and protons

• More effective for higher energy charged particles

More effective for higher energy charged particles More effective for higher energy charged particles More effective for higher energy charged particles

• Started from dusk and midnight sector and propagate to noon sect

Started from dusk and midnight sector and propagate to noon sect Started from dusk and midnight sector and propagate to noon sect Started from dusk and midnight sector and propagate to noon sector

• r
• r
• r

Electron Energy Dispersion Electron Energy Dispersion Electron Energy Dispersion Electron Energy Dispersion

STSAT STSAT STSAT STSAT-

• 1 Data observed on Jun 14, 2004

1 Data observed on Jun 14, 2004 1 Data observed on Jun 14, 2004 1 Data observed on Jun 14, 2004

Can the Tsyganenko model reproduce the energy dispersion?

Jun 14, 2004 Data

• Dst index : -20 nT
• Solar wind pressure : 2.81 nPa
• IMF By : 7.25 nT
• IMF Bz : -7.1 nT

Real Input Parameters

• Dst index : -33 nT
• Solar wind pressure : 2.81 nPa
• IMF By : 7.25 nT
• IMF Bz : -9.1 nT

Electron Orbits on the XY Plan Electron Orbits on the XY Plan Electron Orbits on the XY Plan Electron Orbits on the XY Plan

Waves contribute to loss and acceleration Waves contribute to loss and acceleration Waves contribute to loss and acceleration Waves contribute to loss and acceleration

There were no significant precipitations There were no significant precipitations There were no significant precipitations There were no significant precipitations except mid night sector except mid night sector except mid night sector except mid night sector

E > 300keV

Acceleration Process Acceleration Process Acceleration Process Acceleration Process

Perpendicular Electric Field Perpendicular Electric Field Perpendicular Electric Field Perpendicular Electric Field

Evidence of Local Electron Acceleration

Calculating Radiation Belt Dropout Time

• Electron energy : 150 ~ 315 KeV
• Flux density at the equator (LANL) :

1x10^4 / cm^2 sr s

• Precipitating flux (STSAT-1) : 3x10^5

/ cm^2 sr s

• Magnetic field at the equator : 100 nT
• Magnetic field at 680 km : 40000 nT
• Estimated dropout time = Total

electron number in a flux tube / precipitating flux

~ 2 min ~ 2 min ~ 2 min ~ 2 min

Perpendicular electric field can not accelerate Perpendicular electric field can not accelerate Perpendicular electric field can not accelerate Perpendicular electric field can not accelerate charged particles. charged particles. charged particles. charged particles. Is it true? Is it true? Is it true? Is it true?

Charged Particle Motion in E/M Field Charged Particle Motion in E/M Field Charged Particle Motion in E/M Field Charged Particle Motion in E/M Field (Uniform E field) (Uniform E field) (Uniform E field) (Uniform E field)

• Equation

Equation Equation Equation of Motion

• f Motion
• f Motion
• f Motion
• ExB Drift motion

ExB Drift motion ExB Drift motion ExB Drift motion

• Energy & Charge Independent

Energy & Charge Independent Energy & Charge Independent Energy & Charge Independent

• No energy gain under the condition of

No energy gain under the condition of No energy gain under the condition of No energy gain under the condition of

uniform electric field uniform electric field uniform electric field uniform electric field

• ×

=

• +

× =

• E

E E E (+y) (+y) (+y) (+y) B (+z) B (+z) B (+z) B (+z) (+x) (+x) (+x) (+x)

Electron Trajectory Electron Trajectory Electron Trajectory Electron Trajectory Electron Energy Electron Energy Electron Energy Electron Energy

Charged Particle Motion in E/M Field Charged Particle Motion in E/M Field Charged Particle Motion in E/M Field Charged Particle Motion in E/M Field (Non (Non (Non (Non-

• Uniform E field)

Uniform E field) Uniform E field) Uniform E field)

E = 20 mV/m E = 20 mV/m E = 20 mV/m E = 20 mV/m E = 0 E = 0 E = 0 E = 0

Acceleration Acceleration Acceleration Acceleration By E field By E field By E field By E field Free Gyro Free Gyro Free Gyro Free Gyro-

• motion

motion motion motion B = 20 nT (+z) B = 20 nT (+z) B = 20 nT (+z) B = 20 nT (+z)

Electrons can be accelerated in the non Electrons can be accelerated in the non Electrons can be accelerated in the non Electrons can be accelerated in the non-

• uniform electric field space.

uniform electric field space. uniform electric field space. uniform electric field space. This is very fast acceleration process. This is very fast acceleration process. This is very fast acceleration process. This is very fast acceleration process.

Electron Energy Electron Energy Electron Energy Electron Energy

Electron energy gain from 1MeV to 2 MeV Electron energy gain from 1MeV to 2 MeV Electron energy gain from 1MeV to 2 MeV Electron energy gain from 1MeV to 2 MeV for 40 msec for 40 msec for 40 msec for 40 msec

E E E E B B B B

Electron motion in the electric field gradient Electron motion in the electric field gradient Electron motion in the electric field gradient Electron motion in the electric field gradient and uniform magnetic field and uniform magnetic field and uniform magnetic field and uniform magnetic field

Relativistic Electron Acceleration Relativistic Electron Acceleration Relativistic Electron Acceleration Relativistic Electron Acceleration

E E E Ey

y y y = 70

= 70 = 70 = 70

• 42 mV/m

42 mV/m 42 mV/m 42 mV/m

E E E Ey

y y y Electric Field

Electric Field Electric Field Electric Field Electron Trajectory Electron Trajectory Electron Trajectory Electron Trajectory Electron Energy Electron Energy Electron Energy Electron Energy

Electric field changes from 70 to 42 mV/m in Electric field changes from 70 to 42 mV/m in Electric field changes from 70 to 42 mV/m in Electric field changes from 70 to 42 mV/m in the uniform magnetic field of 20 nT. the uniform magnetic field of 20 nT. the uniform magnetic field of 20 nT. the uniform magnetic field of 20 nT. Electrons drift to +x direction by ExB. Electrons drift to +x direction by ExB. Electrons drift to +x direction by ExB. Electrons drift to +x direction by ExB. Electron gain energy form 1MeV to 1.9 MeV Electron gain energy form 1MeV to 1.9 MeV Electron gain energy form 1MeV to 1.9 MeV Electron gain energy form 1MeV to 1.9 MeV for about 70 msec. for about 70 msec. for about 70 msec. for about 70 msec. Electric field increase makes electron Electric field increase makes electron Electric field increase makes electron Electric field increase makes electron deceleration. deceleration. deceleration. deceleration. Just small change of perpendicular electric Just small change of perpendicular electric Just small change of perpendicular electric Just small change of perpendicular electric fields accelerates or decelerate electrons. fields accelerates or decelerate electrons. fields accelerates or decelerate electrons. fields accelerates or decelerate electrons.

Energetic Electron Injection Energetic Electron Injection Energetic Electron Injection Energetic Electron Injection

Uniform e Uniform e Uniform e Uniform e-

• folding energy

folding energy folding energy folding energy Precipitating electrons Precipitating electrons Precipitating electrons Precipitating electrons ExB drift ExB drift ExB drift ExB drift Energized electrons Energized electrons Energized electrons Energized electrons

Energetic electrons are injected by ExB drift Energetic electrons are injected by ExB drift Energetic electrons are injected by ExB drift Energetic electrons are injected by ExB drift through plasma sheet. through plasma sheet. through plasma sheet. through plasma sheet. Electrons gain energy at the injection boundary Electrons gain energy at the injection boundary Electrons gain energy at the injection boundary Electrons gain energy at the injection boundary where electric field should be decreased. where electric field should be decreased. where electric field should be decreased. where electric field should be decreased. Non Non Non Non-

• uniform E

uniform E uniform E uniform E-

• field acceleration model can

field acceleration model can field acceleration model can field acceleration model can explain trapped particle acceleration explain trapped particle acceleration explain trapped particle acceleration explain trapped particle acceleration.

. . .

Oct 13, 2004 Oct 13, 2004 Oct 13, 2004 Oct 13, 2004 Trapped electrons Trapped electrons Trapped electrons Trapped electrons

Electron Acceleration in the Curvature Field Electron Acceleration in the Curvature Field Electron Acceleration in the Curvature Field Electron Acceleration in the Curvature Field (Uniform E field) (Uniform E field) (Uniform E field) (Uniform E field)

Ey = Ey = Ey = Ey = -

• V x B

V x B V x B V x B

Solar Wind Solar Wind Solar Wind Solar Wind IMF Bz IMF Bz IMF Bz IMF Bz

Several keV electron injection Several keV electron injection Several keV electron injection Several keV electron injection

• Accelerated by Ey

Accelerated by Ey Accelerated by Ey Accelerated by Ey

• Precipitation into atmosphere

Precipitation into atmosphere Precipitation into atmosphere Precipitation into atmosphere

Electrons are accelerated to 1 MeV Electrons are accelerated to 1 MeV Electrons are accelerated to 1 MeV Electrons are accelerated to 1 MeV

Electric Field decreased from 1.3 Re Initial Pitch Angle: 15° Non uniform electric field acceleration is efficient for larger pitch angle electrons

Non Uniform Electric Field Acceleration Non Uniform Electric Field Acceleration Non Uniform Electric Field Acceleration Non Uniform Electric Field Acceleration

Energetic electron precipitation on the polar region Energetic electron precipitation on the polar region Energetic electron precipitation on the polar region Energetic electron precipitation on the polar region

Relativistic Electron Acceleration Relativistic Electron Acceleration Relativistic Electron Acceleration Relativistic Electron Acceleration (in (in (in (in electric field cavity) electric field cavity) electric field cavity) electric field cavity)

Ey Electric Field Ey Electric Field Ey Electric Field Ey Electric Field Electron Trajectory Electron Trajectory Electron Trajectory Electron Trajectory Electron Energy Electron Energy Electron Energy Electron Energy

Electric field change Electric field change Electric field change Electric field change 50 mV/m 50 mV/m 50 mV/m 50 mV/m

• 50 mV/m

50 mV/m 50 mV/m 50 mV/m Cavity size : 0.2 Re Cavity size : 0.2 Re Cavity size : 0.2 Re Cavity size : 0.2 Re Electron energy increased from 1 MeV to 4.6 Electron energy increased from 1 MeV to 4.6 Electron energy increased from 1 MeV to 4.6 Electron energy increased from 1 MeV to 4.6 MeV and decreased to 1.5 MeV. MeV and decreased to 1.5 MeV. MeV and decreased to 1.5 MeV. MeV and decreased to 1.5 MeV. Perpendicular electric fields can make fast Perpendicular electric fields can make fast Perpendicular electric fields can make fast Perpendicular electric fields can make fast charged particle acceleration process. charged particle acceleration process. charged particle acceleration process. charged particle acceleration process.

• Energetic electron precipitation was

Energetic electron precipitation was Energetic electron precipitation was Energetic electron precipitation was enhanced in the scale length of 0.43 Re. enhanced in the scale length of 0.43 Re. enhanced in the scale length of 0.43 Re. enhanced in the scale length of 0.43 Re. The e The e The e The e-

• folding energy increased sharply.

folding energy increased sharply. folding energy increased sharply. folding energy increased sharply. Ionospheric plasma depletion was Ionospheric plasma depletion was Ionospheric plasma depletion was Ionospheric plasma depletion was

• bserved at the same region.
• bserved at the same region.
• bserved at the same region.
• bserved at the same region.

Magnetic field data shows strong upward Magnetic field data shows strong upward Magnetic field data shows strong upward Magnetic field data shows strong upward electric current. electric current. electric current. electric current.

• The electric field might be reduced by

The electric field might be reduced by The electric field might be reduced by The electric field might be reduced by thermal plasma shielding thermal plasma shielding thermal plasma shielding thermal plasma shielding

Energy Diffusion Coefficient by wave particle interaction Energy Diffusion Coefficient by wave particle interaction Energy Diffusion Coefficient by wave particle interaction Energy Diffusion Coefficient by wave particle interaction

Glauert and Horn JGR 2005

Wave Particle Interaction Wave Particle Interaction Wave Particle Interaction Wave Particle Interaction

Electron Microburst Electron Microburst Electron Microburst Electron Microburst

What is the What is the What is the What is the electron micro electron micro electron micro electron micro-

• bursts?

bursts? bursts? bursts?

Strong energetic (~ hundred keV) Strong energetic (~ hundred keV) Strong energetic (~ hundred keV) Strong energetic (~ hundred keV) electron precipitations which duration is electron precipitations which duration is electron precipitations which duration is electron precipitations which duration is less than less than less than less than 1 second. 1 second. 1 second. 1 second.

Electron microburst observed by Electron microburst observed by Electron microburst observed by Electron microburst observed by Korean STSAT Korean STSAT Korean STSAT Korean STSAT-

• 1

1 1 1

1 Second 1 Second 1 Second 1 Second

Electron Electron Electron Electron Microbursts Microbursts Microbursts Microbursts

What causes electron microbursts? What causes electron microbursts? What causes electron microbursts? What causes electron microbursts?

It is presumed that It is presumed that It is presumed that It is presumed that VLF Chorus wave VLF Chorus wave VLF Chorus wave VLF Chorus wave generates generates generates generates the microbursts, because the microbursts, because the microbursts, because the microbursts, because

• a. Both occur preferentially in the morning
• a. Both occur preferentially in the morning
• a. Both occur preferentially in the morning
• a. Both occur preferentially in the morning

sector. sector. sector. sector.

• b. Strong intensity and short duration
• b. Strong intensity and short duration
• b. Strong intensity and short duration
• b. Strong intensity and short duration

(~200msec) of Chorus wave. (~200msec) of Chorus wave. (~200msec) of Chorus wave. (~200msec) of Chorus wave.

• c. Correlation between chorus occurrence and
• c. Correlation between chorus occurrence and
• c. Correlation between chorus occurrence and
• c. Correlation between chorus occurrence and

microburst. microburst. microburst. microburst.

Theoretical diffusion coefficients Theoretical diffusion coefficients Theoretical diffusion coefficients Theoretical diffusion coefficients

• D = 6.0X10-5 rad2/sec (Horn, 2003) Quasi linear

theory

• D = 1.5X10-5 rad2/sec (Albert, 2002) Test particle

simulation

• D = 1.0X10-5 rad2/sec (Inan, 1987) Test particle

simulation with incoherent wave

• Diffusion coefficient should be larger than 3.5 X 10-2

rad2/sec (Lee et al. Geophys. Res. Lett., 2005).

Pitch angle diffusion coefficient by wave Pitch angle diffusion coefficient by wave Pitch angle diffusion coefficient by wave Pitch angle diffusion coefficient by wave particle interaction particle interaction particle interaction particle interaction

Glauert and Horn JGR 2005

Pitch angle diffusion equation Pitch angle diffusion equation Pitch angle diffusion equation Pitch angle diffusion equation

f t 1 sin D sin f

How do electrons interact with waves? How do electrons interact with waves? How do electrons interact with waves? How do electrons interact with waves?

Initial pitch angle : 0 Initial pitch angle : 0 Initial pitch angle : 0 Initial pitch angle : 0 Initial pitch angle : Initial pitch angle : Initial pitch angle : Initial pitch angle : α α α α0

2 4 6

• θ

.

• θ

α α θ α

• +

〈 〈 −

Test Particle Simulation

Method Method Method Method

• a. Solve numerically Lorentz equation of motion for each
• a. Solve numerically Lorentz equation of motion for each
• a. Solve numerically Lorentz equation of motion for each
• a. Solve numerically Lorentz equation of motion for each

electron electron electron electron

• b. Used 720 x 500 test particles ( 0
• b. Used 720 x 500 test particles ( 0
• b. Used 720 x 500 test particles ( 0
• b. Used 720 x 500 test particles ( 0˚

˚ ˚ ˚ ~ 5 ~ 5 ~ 5 ~ 5˚ ˚ ˚ ˚) ) ) )

• c. Input parameters
• c. Input parameters
• c. Input parameters
• c. Input parameters

Electron energy : 300keV Electron energy : 300keV Electron energy : 300keV Electron energy : 300keV Electron cyclotron frequency : f Electron cyclotron frequency : f Electron cyclotron frequency : f Electron cyclotron frequency : fc

c c c = 9.3 kHz (B

= 9.3 kHz (B = 9.3 kHz (B = 9.3 kHz (B0

0 = 334nT)

= 334nT) = 334nT) = 334nT) Electron plasma frequency : f Electron plasma frequency : f Electron plasma frequency : f Electron plasma frequency : fp

p p p = 113 kHz

= 113 kHz = 113 kHz = 113 kHz Wave frequency : Wave frequency : Wave frequency : Wave frequency : ω ω ω ωm

m m m = 1k

= 1k = 1k = 1k × × × × 2 2 2 2π π π π rad/sec rad/sec rad/sec rad/sec δω

δω δω δω = 1k

= 1k = 1k = 1k × × × × 2 2 2 2π

π π π rad/sec

rad/sec rad/sec rad/sec Wave amplitude : B Wave amplitude : B Wave amplitude : B Wave amplitude : Bwave

wave wave wave = 0.1nT

= 0.1nT = 0.1nT = 0.1nT

• !

!

• δω

δω ω

δω ω ω

• −

− = − −

∑ ∫

≈ =

Results Results Results Results

T = 0 msec T = 2 msec T = 4 msec T = 6 msec

Where do electrons interact with waves ? Where do electrons interact with waves ? Where do electrons interact with waves ? Where do electrons interact with waves ?

γ ω

• =
• −
• ω

ω ω ω −

• +

=

• "
• p

8.98 10

3 ne 1 2

B

1 2

Resonance Condition Resonance Condition Resonance Condition Resonance Condition Dispersion Relation Dispersion Relation Dispersion Relation Dispersion Relation Plasma Frequency Plasma Frequency Plasma Frequency Plasma Frequency

θ θ

• #
• +

=

Wave Particle Interaction

Energy Dispersion when electrons diffuse at single point

Energy Dispersion when electrons diffuse at multipoint

Thank You !! Thank You !! Thank You !! Thank You !!