EXPERIMENTS ONTHE RAMSAUER—TOWNSEND EFFECT

& THE CHILD—LANGMUIR LAW

赵元晟, 贾晓瀚

复旦大学物理学系

June 14th, 2017

Contents

• The Ramsauer—Townsend Effect• Introduction & Basic Principle

• Experimental Results

• The Child—Langmuir Law• Basic Principle & Experiment Results

• Correction

• Explanation of the R—T effect by quantum mechanics• Quantum scattering theory

• Analysis of the experimental results

Ramsauer-Townsend Effect?

Principle

• Measuring Scattering Section

Principle

• Measuring Scattering Section

20

21

10

SP

SSS

SK

IIIIIIIII

0

1IIP P

S

Principle

• Geometrical Factor

• 77K

1

0

SII

f 1

11S

PS If

IP

*

*

S

P

IIf *

*

1

1P

S

S

PS I

IIIP

Principle

• Measuring Scattering Section

• f<<1

LnS exP 1)(

*

*

ln1)1ln(1PS

SPs II

IIL

PL

Q

Results

• V~Is,Ip

Results

• V~Is*,Ip*

Results

• Geometrical Factor : f << 1

Results

• V~Q

Results

• V~Ps

Results

• Ionization

The Characteristic Of Thyratrons:The Child–Langmuir Law

A self-consistent process:

∂2Φ

∂z2 =− ρ

ε0, Charge → Electric field;

1

2mv2 = eΦ, Electric field → Current;

J = ρv ,

=⇒ J ∝Va3/2.

Assumption:

Ï Source → infinite; !!!

Ï No initial velocity; !!!

Ï No collision with gas. X

e-

Φ

K S

Filam

ent

0 L = 6mm

Experimental Result

Anomalies

Ï C–L law ⇒ I independent of Vf, but Vf ↑⇒ I ↑?

Ï Slope 6= 1.5? And Vf ↑⇒ Slope ↓?

Vf (V) Slope1.50 1.86†

1.80 1.363.00 1.156.97 1.01†Before saturation.

0.0 0.5 1.0 1.5 2.00

2

4

6

8

ln(Φ (V))

ln(I(μA))

Vf=1.50 V 3.00 V

1.80 V 6.97 V

The Explanation Of The Anomalies:Extension Of The C–L Law

Ï Initial velocity →

eΦ= 1

2m(v2 − v0

2),

Ï Solution:

J ∝ (2eΦ+mv0u)2

u, with u =

√v0

2 + 2eΦ

m− v0.

Ï Log-log-curve → Slightly concave up.

mv02/

(2k),(K) 300 1000 1500 2500ln

(J (1V)

),(A/m2) −2.32 −2.05 −1.93 −1.76

Avg. Slope, (1 ∼ 10V) 1.38 1.31 1.28 1.25

A Further Try

Ï General behavior of Φ(z):Ï “

p” shaped;

Ï zm ≈ 0.

Ï Energy distribution ofthermal electrons:

f (E) ∼ e−E/(kT )

is relatively good.

Ï Self-consistent process(Extended from C–L law):

Φ((f

((ρgg .

Ï Iteration.

z

Φ(z)

Fast e-Slow e-

zm

Φm

ImprovementsÏ Curvature → Larger;

Ï Dependence of J upon Tf

→ Larger;

DeficienciesÏ Too small:

Ï Value of slope;Ï Dependence of J ← Tf.

Ï Field emission neglected.

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

ln(Φ (V))

lnJ(A

/m2)+2

2500 K

1800 K

1200 K

950 K

Someimprovements

But not fullysatisfactory

Explanation Of The Ramsaur–Townsend Effect

Ï Recall: Finite square well potential:

Ï R = 0 when E is the eigen-energy of ISW potential.

V

a

E

R

Quantum Scattering Theory

Ï Schrödinger equation;Ï Boundary condition:

ψ

≈ A

(eikz + f (θ)

eikr

r

), For r →+∞,

is finite, For r → 0.

ⅇⅈ k z

ⅇⅈ k r

r

Core region

V ≠ 0

Radiation region

V ≈ 0

Ï V has SO(3) symmetry → Separate variables.

Ï Radical solution:

R(1)l (r, [V ]), Core region,

R(2)l = Ail (2l +1)

[jl (kr )+ ial h(1)

l (kr )]

, Radiation region.

Ï Connection condition (ρ0 is the radius of core region):

R(2)l (r )

∂r R(2)l (r )

∣∣∣∣∣r=ρ0

=R(1)

l (r )

∂r R(1)l (r )

∣∣∣∣∣r=ρ0

, ⇒ al .

Ï Scattering section:

dσ

dΩ= 1

k2

∣∣∣∣∣∞∑

l=0(2l +1)al Pl (cosθ)

∣∣∣∣∣

2

,

σ= 4π

k2

∞∑

l=0|al |2.

Question: V === ?

Ï Holtsmark: Hartree field & empirical polarization correction.The R–T effect: X; Maximum @ Φ∼ 10V: X.

Ï Allis & Morse: A simplified field: V ∝−maxr−1 − r0−1,0.

The R–T effect: ×; Maximum @ Φ∼ 10V: X.

Atomic Potential (Kr)

Ï Atomic field → Hartree field; Modification → Polarization.Ï Asymptotic behavior of polarization energy:

Φp →αr−4/2, (in atomic units).

Ï The scattering property of Xe is similar.

0 1 2 3 4 5 6 70

2

4

6

8

r (a.u.)

-2Φr2(a.u.) Hartree

& Polarization

Calculation Results

Ï ρ0 = 120a0, α= 16.337a.u.Ï Solve Schrödinger equation of core region numerically.Ï Use connection condition to obtain al .Ï Minimum: Almost exact (!?); Maximum: Left-shifted.

0.5 1.0 1.5 2.0 2.5 3.0 3.50

1

2

3

4

5

Φ1/2 (V1/2)

σ(Rel)

Exp

Tot

s

p

d

f

Correction: Effective Range Of Scattering Angle

Ï Effective range: 0 < θm < θ <π,

σe =∫

θm<θ<πdσ

dΩdΩ.

Ï The position of the maximum→ Largely corrected.

53

S P6mm

2 θm

0.5 1.0 1.5 2.0 2.5 3.0 3.50

1

2

3

4

5

Φ1/2 (V1/2)

σ(Rel)

No Corr

θm=atg(5/12)

atg(5/6)

Exp

Other Corrections

Ï Initial velocity of electrons & Volta potential→ E = e(Φ−Va0) (→ Right?).

Ï Velocity distribution of electrons→ Correction of the curve @ Φ→ 0 (→ Left & Down).

Ï Early scattering (→ Left) or multiple scattering.

Ï Calculated value → empirical polarization.

0 1 2

0

1

∫x

+∞(ζ - 1)2 4 ⅇ-4 (ζ-x) ⅆζ

(x - 1)2

S PK

Ip

Is1

Is2Is3

Is2 and Is3 areindistinguishable!

Without Polarization Energy: No R–T Effect!

Ï ρ0 = 7a0.

Ï Share no similarity with experimental result.→ the polarization term is necessary.

0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

Φ1/2 (V1/2)

σ(Rel)

s f

p Total

d

Discusstion: The Finite Square Well Potential

Ï Wave function: R(1)l (r ) = jl (κr ), with κ=p

2m(E +V )/ħ;

Ï Requirement: 4π3 a3V = 4π

∫ +∞0 r 2VHp(r )dr ;

Ï optimize: a → 164pm;Ï Minimum 6← σ0; Maximum ← σ3.

0 1 2 3 4 5

0

1

2

3

4

5

Φ1/2 (V1/2)

σ(Rel)

s Tot

p f

d

Conclusion

The Ramsauer–Townsend Effect

Ï Min @0.77p

V, Max @2.95p

V; R–T effect verified.Ï Source of errors:

Ï Effective range of scattering angle;Ï Initial velocity distribution of thermal electrons;Ï Volta potential;Ï Early scattering or multiple scattering;Ï Etc.

Ï Problem on FSW potential.

The Child–Langmuir Law

Ï ln I – lnΦ: Slope 6= 1.5; Vf ↑⇒ Slope ↓; Vf ↑⇒ I ↑;Ï Reason:

Ï Initial velocity distribution of thermal electrons;Ï Finite amount of thermal electrons;Ï Volta potential;Ï Field electron emission.Ï Etc.

Thank You For Listening

• 戴道宣, 戴乐山. 近代物理实验. 高等教育出版社. 2006.

• N. F. Mott, H. S. W. Massey. The Theory Of Atomic Collisions. Oxford At The Clarendon Press, 1949.

• H. S. W. Massey, E. H. S. Burhop. Electronic And Ionic Impact Phenomena. Oxford At The Clarendon Press, 1969.

• Yon J. Holtsmark. J., Z. Phys. 55 (1929) 437.

• Yon J. Holtsmark. J., Z. Phys. 66 (1930) 49.

• D. J. Griffiths. Introduction To Quantum Mechanics. Pearson Prentice Hall, 2004.

• 曹钧植, 杨新菊, 姚红英. 物理实验. Vol. 35, No. 3 (2015)

• Thermionic Emission. http://www.physics.csbsju.edu/370/thermionic.pdf

Self-Consistent Field Method

Charge → Electric field:

Φ(z) =∫ L

0

ρ(ζ)

2ε0· |z −ζ|dζ+a +bz,

where a and b are determined by Φ(0) = 0 and Φ(L) =Va .Electric field & Initial velocity → Current:

fz (E) =

f0(E +eΦ(z)

)E > 0,

fz (−E) Φm −Φ(z) < E < 0,

0 E <Φm −Φ(z),

z < zm ,

f0

(E +eΦ(z)

)E >Φ(z)−Φm ,

0 E <Φ(z)−Φm ,z > zm ,

where f = ddE NPassing through.

Initial distribution:

f0(E) = kmT e−W /(kT )

2h3π2 e−E/(kT ).

Relation between ρ and f :

ρ(z) =−e∫ +∞

−∞fz (E)

√m

2|E | dE ,

Similar self-consistent process:

Φ((f

((ρgg .

Iteration!Expression of J :

J = e∫ +∞

−eΦm

f0(E)dE .

Quamtum Scattering Theory

Ï Schrödinger equation;(− ħ2

2m∇2 +V

)ψ= Eψ.

Ï Boundary condition:

ψ

≈ A

(eikz + f (θ)

eikr

r

), For r →+∞,

is finite, For r → 0.

Ï V has SO(3) symmetry → Separate variables.Radical equation:

− ħ2

2m

d2u

dr 2 +[

V + ħ2

2m

l (l +1)

r 2

]u = Eu, with u(r ) ≡ r R(r ).

Ï General solution:

ψ=∞∑

l=0Rl (r )Pl (θ).

Ï Assumption: V negligible for r > ρ0 → Solution:

R(2)l = Ail (2l +1)

[jl (kr )+ ial h(1)

l (kr )]

, For r > ρ0,

Ï Suppose the radical solution for r < ρ0 is R(1)l (r ).

Ï Connection condition:

R(2)l (r )

∂r R(2)l (r )

∣∣∣∣∣r=ρ0

=R(1)

l (r )

∂r R(1)l (r )

∣∣∣∣∣r=ρ0

, ⇒ al .

Ï Scattering section:

dσ

dΩ= 1

k2

∣∣∣∣∣∞∑

l=0(2l +1)al Pl (cosθ)

∣∣∣∣∣

2

,

σ= 4π

k2

∞∑

l=0|al |2.