Nuclear Physics - hep.fcfm.buap.mxhep.fcfm.buap.mx/ecursos/MCII/mec/aplicaciones_d_qm/Nuclear-Ph… · Nuclear Absorption The inverse process of -ray emission is -ray absorption a

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Nuclear Physics(PHY-231)

Dr C. M. Cormack

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Lecture 11 Nuclear Physics Dr C M Cormack 2

Nuclear Physics – This Lecture

This LectureWe will discuss an important effect in nuclear spectroscopy

The Mössbauer Effect – and its applications in technology αααα-ββββ-γγγγ-decay and nuclear spectroscopy

This will be the last in the series of lectures investigating properties of nuclear energy levels before we move on to nuclear technologies and applications to other branches of investigation

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Nuclear AbsorptionThe inverse process of � -ray emission is � -ray absorption a nucleus in its ground state absorbs a photon of energy E� and jumps to an excited state an energy � E above the ground state, these are related by the following equation:

2

2

2Mc

EEE γ

γ −=∆

As the energy of the excited state is not sharp (as the state

is short lived) the absorption will take place when the gamma

energy differs somewhat from the resonant value, the

measurement of the resonant cross section gives the

following

( )( )[ ] ( )22

2

0

2

2)(Γ++∆−

Γ=

REEEE

γ

γ σσ

E∆ γEREE +∆

)( γσ E

Γ

τ

�=Γ

Partial Width


Resonant AbsorptionA schematic of a resonant absorption experiment is shown below:

γE

Inte

nsity

Detector

Nuclei

At energies far from the resonance, the nuclei are transparent to the radiation and no absorption occurs. At resonance the transmitted intensity reaches a minimum value.

In practice it would not be possible to observe the natural line width � . This is because the nuclei are not at rest, they are in constant motion due to thermal excitation. If the motion of the nuclei is represented by the usual Maxwell velocity distribution there will be a distribution of energies of the form ( )( )γγ EEkTmce /12/2 ′−−

2

22ln2

Mc

kTEγ=∆

Doppler Width

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Resonant Absorption 2At room temperature, kT� 0.025 eV and for a 100 keV transition in a medium weight nucleus

�� 0.1 eV, which dominates the natural linewidth for most

nuclear transitions. Even cooling to low temperatures (eg 4K) reduces the linewidth by only an order of magnitude.

To perform absorption experiments it is necessary to have a tunable source of photons. This can only be achieved from a continuous electromagnetic spectrum as obtained from bremsstrahlung or synchrotron radiation.

In most cases in a laboratory absorption experiments a source

with a downward nuclear transition is used to excite an upward

transition. This transition must be within 0.1eV of the desired

resonant energy. An example of such a transition is shown in the

diagram:


Mössbauer EffectThe main problem with the previous example is that it is very difficult to overcome the recoil energy. One method to overcome this is to Doppler shift the source by moving it close to the absorber.

A more successful technique for over coming the recoil problem is a method know as the Mössbauer effect. Developed by Rudolf Mössbauer.

In his original experiment he used a source of 191Ir(E� =129 keV; ER=0.047 eV).

The important point about his experiment was that the emitting and absorbing nuclei were bound in a crystal lattice.

It was known that typical binding energies of an atom in a lattice are 1-10eV. Consequently the recoil energy of typical nuclear transitions are not sufficient to remove an atom from the lattice.

The effect can be thought of in terms of hitting a brick with a cricket bat (the brick will move!) and then hitting the brick whilst it is in a wall (the brick will almost certainly not move!).

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Mössbauer Effect 2Therefore the mass that appears in the expression for the recoil energy becomes the mass of the entire solid, rather than the mass of one atom.

In addition, a certain fraction of the atoms in a lattice are in the vibrationalground state of thermal motion and consequently shows very little thermal Doppler broadening.

The result is very narrow, overlapping emission and absorption lines, each characterised by the natural line-width. This has been nicely demonstrated by Doppler shift experiments where one source was moved relative to the other at low speed. If the speed is such that the Doppler shift is greater than the natural line-width the resonance is destroyed.

For a resonance of line-width 6x10-6 eV, it is found that the necessary speed is about 15 mm/s !


Mössbauer Effect - UsesThe most remarkable thing about the Mössbauer effect is its extreme precision for the measurement of relative energies. In some circumstances it is possible to measure shifts in energy of one part in 1012.

Further details of the statistical mechanics of the Mössbauer effect can be obtained from Krane.

The reason why I have spent so much time on the Mössbauer effect is that it has applications in an enormous variety of areas. Its main use is in determining the properties of nuclei to high precision.

One of its most significant uses however was as a test of Einsteins general theory of relativity, specifically the gravitational red shift.

One of the corner stones of Einsteins theory is the principle of equivalence – according to which the effects of a local uniform gravitational field cannot be distinguished from those of a uniformly accelerated reference frame.

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Mössbauer Effect – Gravitational Redshift

If we were to observe the emission and absorption of radiation in an accelerated reference frame, in which H is the distance between the source and absorber, then the time H/c necessary for radiation to travel from the source to the absorber, the absorber would require a velocity gH/c, where g is the acceleration.

The radiation photons are therefore Doppler shifted according to

2c

gH

c

v

E

E =∆=∆

This amounts to about 1x10-16 per meter in the Earth’s gravitational field



In the original experiment of Pound and Rebka (Phys. Rev. Lett. 4 337 (1960)), 57Fe was used and the 14.4 keV photons were allowed to travel 22.5 m of the Jefferson Physical Laboratory at Harvard. The effect was expected to be of order 2x10-15.

This required a considerable amount of scientific expertise as the sensitivity of 57Fe of roughly 3x10-13. To observe the small shift (about 10-2 of the resonance width) they concentrated on portions of the side of the resonance curve with the greatest slope. To reduce systematic effects – the source and absorber were held at a constant temperature.

Krane

p36

8 fig

10.

28

6



After 4 months of experiments, the result was �

E/E = (4.902±0.041) x10-15

compared with the expected value 4.905 x10-15 .

This experiment represents one of the most precise tests of the General Theory of Relativity


Other uses of the Mössbauer Effect

The primary application of the Mössbauer effect has been in the study of the interaction of nuclei with their physical and chemical environments.

One example is the study of the effect of the penetration of the atomic wave function into the nuclear volume – this is known as the study of hyperfine interactions.

As the nucleus is not a point object, but a distribution of charge this shifts the energy of the atomic electrons by an amount

�E. However from

conservation of energy arguments the energy levels of the nucleus are shifted by an equal but opposite amount. This gives rise to a transition energy change of:

( ) ( )gge EEEEE ∆+−∆+=

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Mössbauer Effect – Isomer ShiftIf the source and absorber in a Mössbauer experiment had the same chemical environment, the resonance would not be affected, but if the source and absorber are different, then the transition energies are slightly different. The effect is shown in the figure below:

The isomer shift. In different materials, the ground state and the excited states show different shifts. The effect on this experiment is to shift the resonance away. In this experiment the energy shift is achieved by Doppler shifting the source – the velocity of the shift is shown in the figure

State Excited

State Ground

0ESE

Eδ

Source

0E

SE

Absorber


Mössbauer – Zeeman EffectAnother kind of hyperfine coupling is the study of the splitting of the nuclear levels in a magnetic field – known as the Zeeman effect.

The equivalent process in atomic physics results in the removal of the spin degeneracy levels (m-degeneracy) of a level of angular momentum I, the field splits into 2I+1 equally spaced sublevels. The atomic effect shifts the energies by 1 part in 104. The nuclear effect however is 1 part in 1012.

This effect is shown in the figure below:

IHmE IM µ=

ShiftIsomer

2

3

2

1

21+

21−

23+

23−

21−

21+

Splitting Dipole Magnetic

Im

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IextNIJextBJJI mBgmBgmAmE µµ −−=

Nuclear Magnetic Resonance µµµµB >> AWhen the external magnetic field is increased the quantum numbers F are no longer good quantum numbers. The large field breaks the coupling of I and J, instead the good quantum numbers are I, mi and J, mJ.

The perturbed energies are:

The high field splitting of our example becomes:

23

2P

3=F

2=F

1=F

0=F

splittingfieldHigh −−

21+

21−

23+


Mössbauer – Electric QuadrupoleThe nuclear quadrupole moment can interact with an electric field gradient to give an electric quadrupole splitting. This splitting is proportional to m2 , thus m and –m are shifted equally in the same direction. The figure below shows the shift with 57Fe

QE∆

23

23

21±

23±

21±

ShiftIsomer Splitting

Quadrupole

Im

I

��

−+−=)12(4

)1(3)(

2

II

IImeqQmE I

IQ

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Mössbauer - Summary1. Mössbauer spectroscopy has been shown to be a very powerful

technique in determining with high precision the energy levels within a nucleus.

2. It has played an important role in determining the effects on these energy levels due to magnetic and electric fields.

3. Its ability to make high precision measurements has allowed the most accurate tests of Einstein’s general theory of relativity.

4. Mössbauer spectroscopy has also found uses in medical diagnosis, in tests for certain blood diseases (by exploiting the presence of Iron in blood cells)

This section wraps up much of the introduction to nuclear physics theories and techniques, the rest of the lecture course will be devoted to applications of these techniques to technology, astro-physics and cosmology. We will return towards the end of the

course to look at nuclear interactions and summarise what we know about fundamental forces in terms of particle physics


Neutron PhysicsThe uncharged member of the nucleon pair, the neutron plays an important role in the study of nuclear forces.

Unaffected by the Coulomb barrier neutrons of even very low energy (~1 eV or less) can penetrate the nucleus and initiate nuclear reactions.

The lack of charge also posses a problem for detection of neutrons, it also posses a problem for energy selection and focussing.

The first experimental observation of the neutron was in 1930 when Botheand Becker bombarded beryllium with � -particles, however the form of the radiation emitted was not understood until 1932 when Chadwick provided the correct hypothesis for the radiation – he is generally credited with being the discoverer of the neutron.

Neutron physics today provides many tests of fundamental theories, including tests of Grand Unified Theories and recently tests of the quantisation of gravity.

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Neutron SourcesFor the remainder of this lecture we will consider some of the applications of neutrons in nuclear reactions and other areas of physics such as crystallography.

Sources

Beams of neutrons can be produced from a variety of nuclear reactions. We cannot accelerate neutrons as we can charged particles, but we can start with high energy neutrons and reduce their energy through collisions with atoms. This process of slowing is called “moderating”. The energy range of neutrons is given below.

o Thermal E � 0.025 eV

o Epithermal E � 1.0 eV

o Slow E � 1.0 keV

o Fast E � 100keV – 10MeV


Neutron Sources 2One of the main sources of neutrons for use in the laboratory comes from � -Beryllium sources, 9Be has a relatively loosely bound neutron. If a typical

� -particle from a radioactive decay (5-6MeV) strikes a 9Be nucleus a neutron can be released:

nCBeHe +→+ 1294

Photoneutron Sources In a process similar to the one above it is possible to use photons to produce neutrons. The advantage of this method is that it is possible to produce nearly monoenergetic sources such a reaction is shown below:

nBeBe +→+ 89γ

Another very common source of neutrons are from Reactors. We will discuss this in later lectures.

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Absorption and Moderation of Neutrons

As a beam of neutrons travels through bulk matter, the intensity will decrease as neutrons are removed from the beam by nuclear interactions. For fast neutrons many interactions are possible (n,p) (n, � ) or (n,2n), but for slow or thermal neutrons the primary cause for their disappearance is capture.

Often the cross sections for these capture reactions are dominated by one or more resonances where the cross section becomes very large. Off resonance, the cross section decreases with increasing velocity like v-1.

The loss in intensity of neutrons traversing a given material of thickness dx, the neutrons will encounter ndx atoms per unit surface area, where n is the number of atoms per unit volume of the material. If � t is the total cross section, the loss in intensity is:

The importance of this topic will be revealed when we come to discuss reactor physics

ndxIdI tσ−= nxteII σ−=�0


Neutron CollisionsFor an elastic collision between a neutron of initial energy E and velocity vwith a target atom of mass A (at rest) gives the following ratio between the final neutron energy E’.

( )2

2

1

cos21

+++=

′A

AA

E

E θ

Where � is the scattering angle in the centre-of-mass frame. For no scattering � =00 the ratio is 1 ! The maximum energy loss occurs for a head on collision

2

min 1

1 ��

+−=��

�� ′A

A

E

E

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Neutron Collisions 2For scattering of neutron energies below 10 MeV and below, the scattering is mostly s-wave and is thus independent of � .

The results of multiple scattering of neutrons in a medium is shown in the figure below.

After many scatterings the neutron distribution

will approach the Maxwellian energy

distribution shown below:


Neutron DetectorsAs neutrons produce no direct ionisation, neutron detectors must be based on detecting the secondary events produced by nuclear reactions such as (n,p), (n,α), (n,γ) or (n, fission).

For slow and thermal neutrons, detectors based on the (n,p) and (n,α) reactions provide a direct means for observing neutrons, from the ionisation signal from the energetic p or α.

10B is commonly used by producing an ionisation chamber, the reaction is:

α+→+ *710 LinB

For thermal neutrons the cross section has a very high value of 3840b and the cross section follows the 1/v law up to about 100 keV (the cross-section is featureless as there are no resonances present).

Neutron detection and absorption has some Important applications in crystallography.

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Neutron Velocity SelectionEarly devices used for determining neutron energies were mechanical devices called selectors (shown below). This were basically rotating shutters made of highly absorbing materials such as Cadmium (Cd).


Neutron Diffraction

Another way of measuring the energy of neutrons in the thermal region is by using crystal diffraction. Thermal neutrons have a de Broglie wavelength of about 0.1 nm, about the same as the spacing between atoms in a crystal lattice. If a beam of thermal neutrons is incident on a crystal it is possible to select wavelengths from interference maxima from the Bragg condition.

θλ sin2dn =

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Nuclear Physics – Next LectureSummary

Next LectureNuclear Fission

1. Theory

2. Applications – Reactors

This concludes much of the technical introduction to nuclear physics.

We have seen that the Mössbauer effect can provide extremely high resolution measurements of nuclear states.

It is also important uses in testing other physical theories such as General Relativity

Use of Neutrons has been discussed, they have many applications in crystallography, but their most significant use is in Nuclear Fission ….

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Nuclear Physics - hep.fcfm.buap.mxhep.fcfm.buap.mx/ecursos/MCII/mec/aplicaciones_d_qm/Nuclear-Ph… · Nuclear Absorption The inverse process of -ray emission is -ray absorption a