PHYS Introduction to Synchrotron Radiation

Transcription

1 PHYS Introduction to Synchrotron Radiation Term: Spring 2015 Meetings: Tuesday & Thursday 17:00-18:15 Location: 204 Stuart Building Instructor: Carlo Segre Office: 166A Life Sciences Phone: segre@iit.edu Book: Web Site: Elements of Modern X-Ray Physics, 2 nd ed., J. Als-Nielsen and D. McMorrow (Wiley, 2011) segre/phys570/15s C. Segre (IIT) PHYS Spring 2015 January 13, / 22

2 Course objectives Understand the means of production of synchrotron x-ray radiation C. Segre (IIT) PHYS Spring 2015 January 13, / 22

3 Course objectives Understand the means of production of synchrotron x-ray radiation Understand the function of various components of a synchrotron beamline C. Segre (IIT) PHYS Spring 2015 January 13, / 22

4 Course objectives Understand the means of production of synchrotron x-ray radiation Understand the function of various components of a synchrotron beamline Be able to perform calculations in support of a synchrotron experiment C. Segre (IIT) PHYS Spring 2015 January 13, / 22

5 Course objectives Understand the means of production of synchrotron x-ray radiation Understand the function of various components of a synchrotron beamline Be able to perform calculations in support of a synchrotron experiment Understand the physics behind a variety of experimental techniques C. Segre (IIT) PHYS Spring 2015 January 13, / 22

6 Course objectives Understand the means of production of synchrotron x-ray radiation Understand the function of various components of a synchrotron beamline Be able to perform calculations in support of a synchrotron experiment Understand the physics behind a variety of experimental techniques Be able to make an oral presentation of a synchrotron radiation research topic C. Segre (IIT) PHYS Spring 2015 January 13, / 22

7 Course objectives Understand the means of production of synchrotron x-ray radiation Understand the function of various components of a synchrotron beamline Be able to perform calculations in support of a synchrotron experiment Understand the physics behind a variety of experimental techniques Be able to make an oral presentation of a synchrotron radiation research topic Be able to write a General User Proposal in the format used by the Advanced Photon Source C. Segre (IIT) PHYS Spring 2015 January 13, / 22

8 Course syllabus Focus on applications of synchrotron radiation C. Segre (IIT) PHYS Spring 2015 January 13, / 22

9 Course syllabus Focus on applications of synchrotron radiation Homework assignments C. Segre (IIT) PHYS Spring 2015 January 13, / 22

10 Course syllabus Focus on applications of synchrotron radiation Homework assignments In-class student presentations on research topics Choose a research article which features a synchrotron technique Timetable will be posted C. Segre (IIT) PHYS Spring 2015 January 13, / 22

11 Course syllabus Focus on applications of synchrotron radiation Homework assignments In-class student presentations on research topics Choose a research article which features a synchrotron technique Timetable will be posted Final project - writing a General User Proposal Start thinking about a suitable project right away Make proposal and get approval before starting C. Segre (IIT) PHYS Spring 2015 January 13, / 22

12 Course syllabus Focus on applications of synchrotron radiation Homework assignments In-class student presentations on research topics Choose a research article which features a synchrotron technique Timetable will be posted Final project - writing a General User Proposal Start thinking about a suitable project right away Make proposal and get approval before starting Visits to Advanced Photon Source (outside class, not required) All students who plan to attend will need to request badges from APS Go to the APS User Portal, and register as a new user. Use MRCAT (Sector 10) as location of experiment Use Carlo Segre as local contact State that your beamtime will be in the first week of March C. Segre (IIT) PHYS Spring 2015 January 13, / 22

13 Course grading 33% Homework assignments C. Segre (IIT) PHYS Spring 2015 January 13, / 22

14 Course grading 33% Homework assignments Weekly or bi-weekly C. Segre (IIT) PHYS Spring 2015 January 13, / 22

15 Course grading 33% Homework assignments Weekly or bi-weekly Due at beginning of class C. Segre (IIT) PHYS Spring 2015 January 13, / 22

16 Course grading 33% Homework assignments Weekly or bi-weekly Due at beginning of class May be turned in via Blackboard C. Segre (IIT) PHYS Spring 2015 January 13, / 22

17 Course grading 33% Homework assignments Weekly or bi-weekly Due at beginning of class May be turned in via Blackboard 33% General User Proposal C. Segre (IIT) PHYS Spring 2015 January 13, / 22

18 Course grading 33% Homework assignments Weekly or bi-weekly Due at beginning of class May be turned in via Blackboard 33% General User Proposal 33% Final Exam Presentation C. Segre (IIT) PHYS Spring 2015 January 13, / 22

19 Course grading 33% Homework assignments Weekly or bi-weekly Due at beginning of class May be turned in via Blackboard 33% General User Proposal 33% Final Exam Presentation Grading scale A 80% to 100% B 65% to 80% C 50% to 65% E 0% to 50% C. Segre (IIT) PHYS Spring 2015 January 13, / 22

20 Topics to be covered (at a minimum) C. Segre (IIT) PHYS Spring 2015 January 13, / 22

21 Topics to be covered (at a minimum) X-rays and their interaction with matter C. Segre (IIT) PHYS Spring 2015 January 13, / 22

22 Topics to be covered (at a minimum) X-rays and their interaction with matter Sources of x-rays C. Segre (IIT) PHYS Spring 2015 January 13, / 22

23 Topics to be covered (at a minimum) X-rays and their interaction with matter Sources of x-rays Refraction and reflection from interfaces C. Segre (IIT) PHYS Spring 2015 January 13, / 22

24 Topics to be covered (at a minimum) X-rays and their interaction with matter Sources of x-rays Refraction and reflection from interfaces Kinematical diffraction C. Segre (IIT) PHYS Spring 2015 January 13, / 22

25 Topics to be covered (at a minimum) X-rays and their interaction with matter Sources of x-rays Refraction and reflection from interfaces Kinematical diffraction Diffraction by perfect crystals C. Segre (IIT) PHYS Spring 2015 January 13, / 22

26 Topics to be covered (at a minimum) X-rays and their interaction with matter Sources of x-rays Refraction and reflection from interfaces Kinematical diffraction Diffraction by perfect crystals Small angle scattering C. Segre (IIT) PHYS Spring 2015 January 13, / 22

27 Topics to be covered (at a minimum) X-rays and their interaction with matter Sources of x-rays Refraction and reflection from interfaces Kinematical diffraction Diffraction by perfect crystals Small angle scattering Photoelectric absorption C. Segre (IIT) PHYS Spring 2015 January 13, / 22

28 Topics to be covered (at a minimum) X-rays and their interaction with matter Sources of x-rays Refraction and reflection from interfaces Kinematical diffraction Diffraction by perfect crystals Small angle scattering Photoelectric absorption Resonant scattering C. Segre (IIT) PHYS Spring 2015 January 13, / 22

29 Topics to be covered (at a minimum) X-rays and their interaction with matter Sources of x-rays Refraction and reflection from interfaces Kinematical diffraction Diffraction by perfect crystals Small angle scattering Photoelectric absorption Resonant scattering Imaging C. Segre (IIT) PHYS Spring 2015 January 13, / 22

30 Resources for the course Orange x-ray data booklet: C. Segre (IIT) PHYS Spring 2015 January 13, / 22

31 Resources for the course Orange x-ray data booklet: Center for X-Ray Optics web site: C. Segre (IIT) PHYS Spring 2015 January 13, / 22

32 Resources for the course Orange x-ray data booklet: Center for X-Ray Optics web site: Hephaestus from the Demeter suite: C. Segre (IIT) PHYS Spring 2015 January 13, / 22

33 Resources for the course Orange x-ray data booklet: Center for X-Ray Optics web site: Hephaestus from the Demeter suite: McMaster data on the Web: C. Segre (IIT) PHYS Spring 2015 January 13, / 22

34 Resources for the course Orange x-ray data booklet: Center for X-Ray Optics web site: Hephaestus from the Demeter suite: McMaster data on the Web: X-ray Oriented Programs: C. Segre (IIT) PHYS Spring 2015 January 13, / 22

35 Today s outline - January 13, 2015 C. Segre (IIT) PHYS Spring 2015 January 13, / 22

36 Today s outline - January 13, 2015 History of x-ray sources C. Segre (IIT) PHYS Spring 2015 January 13, / 22

37 Today s outline - January 13, 2015 History of x-ray sources X-ray interactions with matter C. Segre (IIT) PHYS Spring 2015 January 13, / 22

38 Today s outline - January 13, 2015 History of x-ray sources X-ray interactions with matter Thomson scattering C. Segre (IIT) PHYS Spring 2015 January 13, / 22

39 Today s outline - January 13, 2015 History of x-ray sources X-ray interactions with matter Thomson scattering Atomic form factor C. Segre (IIT) PHYS Spring 2015 January 13, / 22

40 Today s outline - January 13, 2015 History of x-ray sources X-ray interactions with matter Thomson scattering Atomic form factor Reading Assignment: Chapter ; C. Segre (IIT) PHYS Spring 2015 January 13, / 22

41 History of x-ray sources C. Segre (IIT) PHYS Spring 2015 January 13, / 22

42 History of x-ray sources 1895 x-rays discovered by William Röntgen C. Segre (IIT) PHYS Spring 2015 January 13, / 22

43 History of x-ray sources 1895 x-rays discovered by William Röntgen 1 st generation synchrotrons initially used in parasitic mode (SSRL, CHESS) C. Segre (IIT) PHYS Spring 2015 January 13, / 22

44 History of x-ray sources 1895 x-rays discovered by William Röntgen 1 st generation synchrotrons initially used in parasitic mode (SSRL, CHESS) 2 nd generation were dedicated sources (NSLS, SRC, CAMD) C. Segre (IIT) PHYS Spring 2015 January 13, / 22

45 History of x-ray sources 1895 x-rays discovered by William Röntgen 1 st generation synchrotrons initially used in parasitic mode (SSRL, CHESS) 2 nd generation were dedicated sources (NSLS, SRC, CAMD) 3 rd generation featured insertion devices (APS, ESRF, ALS) C. Segre (IIT) PHYS Spring 2015 January 13, / 22

46 History of x-ray sources 1895 x-rays discovered by William Röntgen 1 st generation synchrotrons initially used in parasitic mode (SSRL, CHESS) 2 nd generation were dedicated sources (NSLS, SRC, CAMD) 3 rd generation featured insertion devices (APS, ESRF, ALS) 4 th generation are free electron lasers (LCLS) C. Segre (IIT) PHYS Spring 2015 January 13, / 22

47 The classical x-ray The classical plane wave representation of x-rays is: C. Segre (IIT) PHYS Spring 2015 January 13, / 22

48 The classical x-ray The classical plane wave representation of x-rays is: E(r, t) = ˆɛE o e i(k r ωt) C. Segre (IIT) PHYS Spring 2015 January 13, / 22

49 The classical x-ray The classical plane wave representation of x-rays is: E(r, t) = ˆɛE o e i(k r ωt) where ˆɛ is a unit vector in the direction of the electric field C. Segre (IIT) PHYS Spring 2015 January 13, / 22

50 The classical x-ray The classical plane wave representation of x-rays is: E(r, t) = ˆɛE o e i(k r ωt) where ˆɛ is a unit vector in the direction of the electric field, k is the wavevector of the radiation along the propagation direction C. Segre (IIT) PHYS Spring 2015 January 13, / 22

51 The classical x-ray The classical plane wave representation of x-rays is: E(r, t) = ˆɛE o e i(k r ωt) where ˆɛ is a unit vector in the direction of the electric field, k is the wavevector of the radiation along the propagation direction, and ω is the angular frequency of oscillation of the radiation. C. Segre (IIT) PHYS Spring 2015 January 13, / 22

52 The classical x-ray The classical plane wave representation of x-rays is: E(r, t) = ˆɛE o e i(k r ωt) where ˆɛ is a unit vector in the direction of the electric field, k is the wavevector of the radiation along the propagation direction, and ω is the angular frequency of oscillation of the radiation. If the energy, E is in kev, the relationship among these quantities is given by: C. Segre (IIT) PHYS Spring 2015 January 13, / 22

53 The classical x-ray The classical plane wave representation of x-rays is: E(r, t) = ˆɛE o e i(k r ωt) where ˆɛ is a unit vector in the direction of the electric field, k is the wavevector of the radiation along the propagation direction, and ω is the angular frequency of oscillation of the radiation. If the energy, E is in kev, the relationship among these quantities is given by: ω = hν = E, λν = c C. Segre (IIT) PHYS Spring 2015 January 13, / 22

54 The classical x-ray The classical plane wave representation of x-rays is: E(r, t) = ˆɛE o e i(k r ωt) where ˆɛ is a unit vector in the direction of the electric field, k is the wavevector of the radiation along the propagation direction, and ω is the angular frequency of oscillation of the radiation. If the energy, E is in kev, the relationship among these quantities is given by: ω = hν = E, λν = c λ = hc/e = ( ev s)( m/s)/e = ( kev s)( Å/s)/E = Å kev/e to give units of Å C. Segre (IIT) PHYS Spring 2015 January 13, / 22

55 Interactions of x-rays with matter For the purposes of this course, we care most about the interactions of x-rays with matter. There are four basic types of such interactions: C. Segre (IIT) PHYS Spring 2015 January 13, / 22

56 Interactions of x-rays with matter For the purposes of this course, we care most about the interactions of x-rays with matter. There are four basic types of such interactions: 1 Elastic scattering C. Segre (IIT) PHYS Spring 2015 January 13, / 22

57 Interactions of x-rays with matter For the purposes of this course, we care most about the interactions of x-rays with matter. There are four basic types of such interactions: 1 Elastic scattering 2 Inelastic scattering C. Segre (IIT) PHYS Spring 2015 January 13, / 22

58 Interactions of x-rays with matter For the purposes of this course, we care most about the interactions of x-rays with matter. There are four basic types of such interactions: 1 Elastic scattering 2 Inelastic scattering 3 Absorption C. Segre (IIT) PHYS Spring 2015 January 13, / 22

59 Interactions of x-rays with matter For the purposes of this course, we care most about the interactions of x-rays with matter. There are four basic types of such interactions: 1 Elastic scattering 2 Inelastic scattering 3 Absorption 4 Pair production C. Segre (IIT) PHYS Spring 2015 January 13, / 22

60 Interactions of x-rays with matter For the purposes of this course, we care most about the interactions of x-rays with matter. There are four basic types of such interactions: 1 Elastic scattering 2 Inelastic scattering 3 Absorption 4 Pair production We will only discuss the first three. C. Segre (IIT) PHYS Spring 2015 January 13, / 22

61 Elastic scattering geometry k Q an incident x-ray of wave number k k C. Segre (IIT) PHYS Spring 2015 January 13, / 22

62 Elastic scattering geometry k Q an incident x-ray of wave number k scatters elastically to k k C. Segre (IIT) PHYS Spring 2015 January 13, / 22

63 Elastic scattering geometry k Q an incident x-ray of wave number k scatters elastically to k resulting in a scattering vector Q k C. Segre (IIT) PHYS Spring 2015 January 13, / 22

64 Elastic scattering geometry k Q an incident x-ray of wave number k scatters elastically to k resulting in a scattering vector Q or in terms of momentum transfer: Q = k k k C. Segre (IIT) PHYS Spring 2015 January 13, / 22

65 Thomson scattering C. Segre (IIT) PHYS Spring 2015 January 13, / 22

66 Thomson scattering Assumptions: plane wave of x-rays incident on a single electron C. Segre (IIT) PHYS Spring 2015 January 13, / 22

67 Thomson scattering Assumptions: plane wave of x-rays incident on a single electron total scattered energy total incoming energy C. Segre (IIT) PHYS Spring 2015 January 13, / 22

68 Thomson scattering Assumptions: plane wave of x-rays incident on a single electron total scattered energy total incoming energy electron is a point charge C. Segre (IIT) PHYS Spring 2015 January 13, / 22

69 Thomson scattering Assumptions: plane wave of x-rays incident on a single electron total scattered energy total incoming energy electron is a point charge scattered intensity 1/R 2 C. Segre (IIT) PHYS Spring 2015 January 13, / 22

70 Thomson scattering E rad (R, t) = e 4πɛ 0 c 2 R a x(t ) sin Ψ C. Segre (IIT) PHYS Spring 2015 January 13, / 22

71 Thomson scattering E rad (R, t) = e 4πɛ 0 c 2 R a x(t ) sin Ψ where t = t R/c C. Segre (IIT) PHYS Spring 2015 January 13, / 22

72 Thomson scattering E rad (R, t) = e 4πɛ 0 c 2 R a x(t ) sin Ψ where t = t R/c a x (t ) = e m E xoe iωt C. Segre (IIT) PHYS Spring 2015 January 13, / 22

73 Thomson scattering E rad (R, t) = e 4πɛ 0 c 2 R a x(t ) sin Ψ where t = t R/c a x (t ) = e m E xoe iωt = e m E xoe iωt e iωr/c C. Segre (IIT) PHYS Spring 2015 January 13, / 22

74 Thomson scattering E rad (R, t) = e 4πɛ 0 c 2 R a x(t ) sin Ψ where t = t R/c a x (t ) = e m E xoe iωt = e m E xoe iωt e iωr/c a x (t ) = e m E ine iωr/c C. Segre (IIT) PHYS Spring 2015 January 13, / 22

75 Thomson scattering E rad (R, t) = e 4πɛ 0 c 2 R e m E ine iωr/c sin Ψ C. Segre (IIT) PHYS Spring 2015 January 13, / 22

76 Thomson scattering E rad (R, t) = e e 4πɛ 0 c 2 R m E ine iωr/c sin Ψ E rad (R, t) = e2 e iωr/c E in 4πɛ 0 mc 2 sin Ψ R C. Segre (IIT) PHYS Spring 2015 January 13, / 22

77 Thomson scattering E rad (R, t) = e e 4πɛ 0 c 2 R m E ine iωr/c sin Ψ E rad (R, t) = e2 e iωr/c E in 4πɛ 0 mc 2 sin Ψ but k = ω/c R C. Segre (IIT) PHYS Spring 2015 January 13, / 22

78 Thomson scattering E rad (R, t) = e2 e ikr E in 4πɛ 0 mc 2 R sin Ψ = r e ikr o R sin Ψ C. Segre (IIT) PHYS Spring 2015 January 13, / 22

79 Thomson scattering E rad (R, t) = e2 e ikr E in 4πɛ 0 mc 2 R r 0 = e 2 4πɛ 0 mc 2 = Å sin Ψ = r e ikr o R sin Ψ C. Segre (IIT) PHYS Spring 2015 January 13, / 22

80 Scattering cross-section C. Segre (IIT) PHYS Spring 2015 January 13, / 22

81 Scattering cross-section Detector of solid angle Ω at a distance R from electron C. Segre (IIT) PHYS Spring 2015 January 13, / 22

82 Scattering cross-section Detector of solid angle Ω at a distance R from electron Cross-section of incoming beam = A o C. Segre (IIT) PHYS Spring 2015 January 13, / 22

83 Scattering cross-section Detector of solid angle Ω at a distance R from electron Cross-section of incoming beam = A o Cross section of scattered beam (into detector) = R 2 Ω C. Segre (IIT) PHYS Spring 2015 January 13, / 22

84 Scattering cross-section Detector of solid angle Ω at a distance R from electron Cross-section of incoming beam = A o Cross section of scattered beam (into detector) = R 2 Ω I scatt I o = E rad 2 E in 2 R2 Ω C. Segre (IIT) PHYS Spring 2015 January 13, / 22

85 Scattering cross-section Differential cross-section is obtained by normalizing C. Segre (IIT) PHYS Spring 2015 January 13, / 22

86 Scattering cross-section Differential cross-section is obtained by normalizing dσ dω = C. Segre (IIT) PHYS Spring 2015 January 13, / 22

87 Scattering cross-section Differential cross-section is obtained by normalizing dσ dω = I scatt (I o /A o ) Ω C. Segre (IIT) PHYS Spring 2015 January 13, / 22

88 Scattering cross-section Differential cross-section is obtained by normalizing dσ dω = I scatt (I o /A o ) Ω = E rad 2 E in 2 R2 C. Segre (IIT) PHYS Spring 2015 January 13, / 22

89 Scattering cross-section Differential cross-section is obtained by normalizing dσ dω = I scatt (I o /A o ) Ω = E rad 2 E in 2 R2 E rad 2 E in 2 R2 = ro 2 e ikr e ikr R 2 R 2 sin 2 Ψ C. Segre (IIT) PHYS Spring 2015 January 13, / 22

90 Scattering cross-section Differential cross-section is obtained by normalizing dσ dω = I scatt (I o /A o ) Ω = E rad 2 E in 2 R2 E rad 2 E in 2 R2 = ro 2 e ikr e ikr R 2 R 2 sin 2 Ψ = ro 2 sin 2 Ψ C. Segre (IIT) PHYS Spring 2015 January 13, / 22

91 Scattering cross-section Differential cross-section is obtained by normalizing dσ dω = I scatt (I o /A o ) Ω = E rad 2 E in 2 R2 = ro 2 sin 2 Ψ E rad 2 E in 2 R2 = ro 2 e ikr e ikr R 2 R 2 sin 2 Ψ = ro 2 sin 2 Ψ C. Segre (IIT) PHYS Spring 2015 January 13, / 22

92 Total cross-section Integrate to obtain the total Thomson scattering cross-section from an electron. C. Segre (IIT) PHYS Spring 2015 January 13, / 22

93 Total cross-section Integrate to obtain the total Thomson scattering cross-section from an electron. σ = 8π 3 r 2 o C. Segre (IIT) PHYS Spring 2015 January 13, / 22

94 Total cross-section Integrate to obtain the total Thomson scattering cross-section from an electron. σ = 8π 3 r o 2 = cm 2 = barn C. Segre (IIT) PHYS Spring 2015 January 13, / 22

95 Total cross-section Integrate to obtain the total Thomson scattering cross-section from an electron. If displacement is in vertical direction, sin Ψ term is replaced by unity and if the source is unpolarized, it is a combination. σ = 8π 3 r o 2 = cm 2 = barn C. Segre (IIT) PHYS Spring 2015 January 13, / 22

96 Total cross-section Integrate to obtain the total Thomson scattering cross-section from an electron. If displacement is in vertical direction, sin Ψ term is replaced by unity and if the source is unpolarized, it is a combination. σ = 8π 3 r o 2 1 Polarization factor = sin 2 Ψ = cm 2 = barn 1 2 ( 1 + sin 2 Ψ ) C. Segre (IIT) PHYS Spring 2015 January 13, / 22

97 Atomic scattering phase shift arises from scattering off different portions of extended electron distribution r θ φ(r) = (k k ) r = Q r C. Segre (IIT) PHYS Spring 2015 January 13, / 22

98 Atomic scattering phase shift arises from scattering off different portions of extended electron distribution r θ φ(r) = (k k ) r = Q r k θ Q Q = 2 k sinθ = 4π λ sinθ k C. Segre (IIT) PHYS Spring 2015 January 13, / 22

99 Atomic form factor the volume element at r contributes r o ρ(r)d 3 r with phase factor e iq r C. Segre (IIT) PHYS Spring 2015 January 13, / 22

100 Atomic form factor the volume element at r contributes r o ρ(r)d 3 r with phase factor e iq r for an entire atom, integrate to get the atomic form factor f o (Q): C. Segre (IIT) PHYS Spring 2015 January 13, / 22

101 Atomic form factor the volume element at r contributes r o ρ(r)d 3 r with phase factor e iq r for an entire atom, integrate to get the atomic form factor f o (Q): r o f o (Q) = r o ρ(r)e iq r d 3 r C. Segre (IIT) PHYS Spring 2015 January 13, / 22

102 Atomic form factor the volume element at r contributes r o ρ(r)d 3 r with phase factor e iq r for an entire atom, integrate to get the atomic form factor f o (Q): r o f o (Q) = r o ρ(r)e iq r d 3 r Electrons which are tightly bound cannot respond like a free electron. This results in a depression of the atomic form factor, called f and a lossy term near an ionization energy, called f. Together these are the anomalous corrections to the atomic form factor. C. Segre (IIT) PHYS Spring 2015 January 13, / 22

103 Atomic form factor the volume element at r contributes r o ρ(r)d 3 r with phase factor e iq r for an entire atom, integrate to get the atomic form factor f o (Q): r o f o (Q) = r o ρ(r)e iq r d 3 r Electrons which are tightly bound cannot respond like a free electron. This results in a depression of the atomic form factor, called f and a lossy term near an ionization energy, called f. Together these are the anomalous corrections to the atomic form factor. f f Energy (ev) C. Segre (IIT) PHYS Spring 2015 January 13, / 22

104 Atomic form factor the volume element at r contributes r o ρ(r)d 3 r with phase factor e iq r for an entire atom, integrate to get the atomic form factor f o (Q): r o f o (Q) = r o ρ(r)e iq r d 3 r Electrons which are tightly bound cannot respond like a free electron. This results in a depression of the atomic form factor, called f and a lossy term near an ionization energy, called f. Together these are the anomalous corrections to the atomic form factor. f the total atomic scattering factor is f Energy (ev) C. Segre (IIT) PHYS Spring 2015 January 13, / 22

105 Atomic form factor the volume element at r contributes r o ρ(r)d 3 r with phase factor e iq r for an entire atom, integrate to get the atomic form factor f o (Q): r o f o (Q) = r o ρ(r)e iq r d 3 r Electrons which are tightly bound cannot respond like a free electron. This results in a depression of the atomic form factor, called f and a lossy term near an ionization energy, called f. Together these are the anomalous corrections to the atomic form factor. f the total atomic scattering factor is f (Q, ω) = f o (Q) + f ( ω) + if ( ω) f Energy (ev) C. Segre (IIT) PHYS Spring 2015 January 13, / 22

106 Atomic form factor The atomic form factor has an angular dependence C. Segre (IIT) PHYS Spring 2015 January 13, / 22

107 Atomic form factor The atomic form factor has an angular dependence Q = 4π λ sin θ C. Segre (IIT) PHYS Spring 2015 January 13, / 22

108 Atomic form factor The atomic form factor has an angular dependence Q = 4π λ sin θ Lighter atoms (blue is oxygen) have wider form factor C. Segre (IIT) PHYS Spring 2015 January 13, / 22

109 Scattering from atoms: all effects Scattering from an atom is built up from component quantities: C. Segre (IIT) PHYS Spring 2015 January 13, / 22

110 Scattering from atoms: all effects Scattering from an atom is built up from component quantities: Thomson scattering from a single electron r o = e2 4πɛ 0 mc 2 r o = r o C. Segre (IIT) PHYS Spring 2015 January 13, / 22

111 Scattering from atoms: all effects Scattering from an atom is built up from component quantities: Thomson scattering from a single electron e2 r o = 4πɛ 0 mc 2 atomic form factor f o (Q) = ρ(r)e iq r d 3 r r o f (Q, ω) = r o [ f o (Q) ] C. Segre (IIT) PHYS Spring 2015 January 13, / 22

112 Scattering from atoms: all effects Scattering from an atom is built up from component quantities: Thomson scattering from a single electron e2 r o = 4πɛ 0 mc 2 atomic form factor f o (Q) = ρ(r)e iq r d 3 r anomalous scattering terms f ( ω) + if ( ω) r o f (Q, ω) = r o [ f o (Q) + f ( ω) + if ( ω) ] C. Segre (IIT) PHYS Spring 2015 January 13, / 22

113 Scattering from atoms: all effects Scattering from an atom is built up from component quantities: Thomson scattering from a single electron e2 r o = 4πɛ 0 mc 2 atomic form factor f o (Q) = ρ(r)e iq r d 3 r anomalous scattering terms polarization factor f ( ω) + if ( ω) 1 P = sin 2 Ψ 1 2 (1 + sin2 Ψ) r o f (Q, ω) sin 2 Ψ = r o [ f o (Q) + f ( ω) + if ( ω) ] sin 2 Ψ C. Segre (IIT) PHYS Spring 2015 January 13, / 22