# 2. rutherford backscattering spectrometry basic principles

Post on 05-Jan-2016

219 views

Category:

## Documents

Tags:

• #### stationary nucleus of

Embed Size (px)

TRANSCRIPT

• 2. RUTHERFORDBACKSCATTERINGSPECTROMETRYBasic Principles

• ELASTIC COULOMB SCATTERINGConsider an ion of charge z, mass m and initial kinetic energy Eo being scattered elastically from a stationary nucleus of charge Z and mass M purely by the Coulomb force. The final energy E of the scattered ion is a function of the angle of scatter from the initial direction and the ratio M/m. m, Eom, EM, 0M, (Eo E)Recoiling nucleusScattered ion

• KINEMATIC FACTORThe scattered E can be derived from the principles of conservation of energy and momentum, and is given by (in laboratory frame of reference):

The multiplication factor k on the right hand side of the equation is often referred to as the kinematic factor. For M/m > 1, k is a slow-varying function of , having the maximum value of 1 at = 0 and the minimum value at = 180o. For M/m = 1, the value of k is zero beyond 90o.

• M/m = 1M/m = 4M/m = 20Plots of k vs for M/m = 0, 4, 20

Chart1

111

0.96984636080.99242910260.9984819026

0.88302241110.97024221740.9939869776

0.7500003830.93495464590.9866897931

0.58682466960.88889450680.9768698698

0.41317663710.83493908020.9648957215

0.2500007660.7762052980.9512047015

0.11697844180.7157420510.9362804818

0.0301540930.65626965980.9206300589

00.60000041110.904762025

00.54855573460.8891674898

00.50297514480.874304575

00.46379537040.8605868866

00.43116962820.8483758945

00.40499779030.8379767554

00.38504567290.8296368511

00.37104153740.8235461802

00.36274639880.8198387371

00.360.8185941043

q

k

Sheet1

14201

angleE1E2E3

01.001.001.00

100.970.991.00

200.880.970.99

300.750.930.99

400.590.890.98

500.410.830.96

600.250.780.95

700.120.720.94

800.030.660.92

900.000.600.90

1000.000.550.89

1100.000.500.87

1200.000.460.86

1300.000.430.85

1400.000.400.84

1500.000.390.83

1600.000.370.82

1700.000.360.82

1800.000.360.82

M/menergy

10.00

50.44

100.67

150.77

200.82

250.85

300.88

350.89

400.90

450.91

500.92

550.93

600.94

650.94

700.94

750.95

800.95

850.95

900.96

950.96

1000.96

Sheet1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Sheet2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Sheet3

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

q

k

• Plots of k vs M/m for = 180oThe small gradient at large M/m ratios implies that mass separation is poor for higher M/m ratios.

Chart2

0

0.4444444444

0.6694214876

0.765625

0.8185941043

0.8520710059

0.8751300728

0.8919753086

0.9048185604

0.9149338374

0.9231064975

0.9298469388

0.9355012094

0.940312213

0.9444554652

0.9480609418

0.9512269471

0.954029205

0.9565269895

0.9587673611

0.960788158

M/m

k

Sheet1

14201

angleE1E2E3

01.001.001.00

100.970.991.00

200.880.970.99

300.750.930.99

400.590.890.98

500.410.830.96

600.250.780.95

700.120.720.94

800.030.660.92

900.000.600.90

1000.000.550.89

1100.000.500.87

1200.000.460.86

1300.000.430.85

1400.000.400.84

1500.000.390.83

1600.000.370.82

1700.000.360.82

1800.000.360.82

M/menergy

10.00

50.44

100.67

150.77

200.82

250.85

300.88

350.89

400.90

450.91

500.92

550.93

600.94

650.94

700.94

750.95

800.95

850.95

900.96

950.96

1000.96

Sheet1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Sheet2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

M/m

k

Sheet3

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

q

k

• DIFFERENTIAL SCATTERING CROSS SECTIONThe theoretical differential cross section for a given scattering angle is given by:

The higher order terms are usually negligible for M>m and the differential cross section can be expressed in the following form, which has the unit of mb/sr (millibarns per steradian) when MeV is used as the unit for Eo:

• ds/d versus qThe dependence of ds/d on sin-4(q/2) means that ds/d has very large values at small forward scattered angles and approaches infinity at 0o. However, for q = 90o to 180o, it decreases rather slowly because sin-4(q/2) drops gradually from a value of 4 at 90o to 1 at 180o. The diagram below shows the plots of ds/d vs q for the Coulomb scattering of 2 MeV 4He+ ions for oxygen and calcium.Ca (Z=20, A=40)O (Z=8, A=16)

Chart2

359367.9002199652246062.98437478

22803.2187742154142533.725338846

4618.466809535928879.0255595992

1512.78137638219468.491602388

647.43787883144060.0947426965

329.18501659742071.014353734

188.99280394031194.813024627

118.8744161849756.5731011557

80.3522200998515.8093756237

57.6237628227373.7565176419

43.4620568532285.2458553327

34.2720753035227.808470647

28.1422708673189.4971929207

24.0019207917163.6200049481

21.2284733005146.2859581281

19.4534774789135.192234243

18.4626618392128.999636495

18.1440000001127.0080000005

qo

ds/dW (mb/sr)

Sheet1

14201

angleE1E2E3

01.001.001.00

100.970.991.00

200.880.970.99

300.750.930.99

400.590.890.98

500.410.830.96

600.250.780.95

700.120.720.94

800.030.660.92

900.000.600.90

1000.000.550.89

1100.000.500.87

1200.000.460.86

1300.000.430.85

1400.000.400.84

1500.000.390.83

1600.000.370.82

1700.000.360.82

1800.000.360.82

M/mk

10.00

50.44

100.67

150.77

200.82

250.85

300.88

350.89

400.90

450.91

500.92

550.93

600.94

650.94

700.94

750.95

800.95

850.95

900.96

950.96

1000.96

Sheet1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Sheet2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

M/m

k

Sheet3

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

q

k

820

0.250.1

qx-sectionx-section

103593682246063

2022803142534

30461828879

4015139468

506474060

603292071

701891195

80119757

9080516

10058374

11043285

12034228

13028189

14024164

15021146

16019135

17018129

18018127

Ex-section

0.17258

0.21814

0.3806

0.4454

0.5290

0.6202

0.7148

0.8113

0.990

1.073

1.160

1.250

1.343

1.437

1.532

1.628

1.725

1.822

1.920

2.018

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

qo

ds/dW (mb/sr)

00

00

00

00

00

00

00

00

00

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

E (MeV)

ds/dW(mb/sr)

• ds/d versus Eods/d is inversely proportional to the square of the incident ion energy Eo, implying that it increases with decreasing Eo. This also means that for a thick target, more ions are scattered from a greater depth. The diagram below show a plot of ds/d vs Eo for the scattering of 4He+ from an aluminium target at q = 180o.Projectile: 4He+Target: AlScatter angle: 180o

Chart4

20942.8996608

5235.7249152

2326.9888512

1308.9312288

837.715986432

581.7472128

427.4061155265

327.2328072

258.5543168

209.428996608

173.0818153785

145.4368032

123.9224832

106.8515288816

93.079554048

81.8082018

72.4667808332

64.6385792

58.0135724676

52.357249152

E (MeV)

ds/dW(mb/sr)

Sheet1

14201

angleE1E2E3

01.001.001.00

100.970.991.00

200.880.970.99

300.750.930.99

400.590.890.98

500.410.830.96

600.250.780.95

700.120.720.94

800.030.660.92

900.000.600.90

1000.000.550.89

1100.000.500.87

1200.000.460.86

1300.000.430.85

1400.000.400.84

1500.000.390.83

1600.000.370.82

1700.

Recommended