4 Basic nuclear properties-6

Sanjay Kumar Chamoli

Learning Outcomes

From this module students may get to know about the following:

The knowledge of basic nuclear properties.
The importance of nuclear properties.
The experimental ways of determining nuclear properties.

1. Measuring nuclear properties; Electromagnetic moments

Electromagnetic moment is one of the basic observables characterizing a particular nucleus. Nuclear moments are crucial ingredients get a clear understanding of the single-particle or collective nature of nuclear states. The magnetic moment is sensitive to the single-particle nature of the valence nucleon, while the nuclear quadrupole moment is sensitive to the deformation. In order to measure the nuclear moments of exotic nuclei in their nuclear states, it is necessary to apply complementary experimental techniques that cover a wide range of nuclear lifetimes and spins. Various experimental techniques to measure the nuclear moments with a particular range of nuclear lifetime is shown in figure 1.

Fig. 1: Experimental techniques to measure the nuclear moments with a particular range of nuclear lifetime.

The Zeeman effect is the effect of splitting a spectral line into several components in the presence of a static magnetic field. Figure 2 shows the hyperfine levels of a nucleus immersed into a static magnetic field. In the figure it is clear that the Zeeman splitting is equidistant and proportional to the Larmor frequency νL. Figure 3 shows the hyperfine levels of a nucleus implanted in a crystal with an electric field gradient. In the figure it is clear that the nuclear level splitting is not equidistant, and proportional to the quadrupole frequency νQ. The interaction between the magnetic moment, due to the spin of the nucleus, and the larger magnetic moment, due to the electron’s spin, results in energy shifts which are hyperfine splitting. On the other hand the impact of the interaction of external electric field with the quadrupole moment of nuclei is frequency dependent. The splitting of nuclear energy levels due to the interaction of an external/internal magnetic field and the external/internal electric field are shown in Fig. 2 & Fig. 3 respectively.

1.1. Magnetic dipole moment

The magnetic dipole moment, μI is defined as the expectation value of the z-component of the dipole operator. i.e.,

1.1.1 Measuring magnetic moment in nuclei : Experimental techniques based on measuring the angular distribution of the radioactive decay give more precise measurements of the nuclear g-factor and quadrupole moment. It is influenced by the interaction of the nuclear moments with externally applied magnetic fields and/or electric field gradients. The radioactive decay intensity is measured as a function of time or as a function of an external variable, e.g., a static magnetic field.

1.1.1.1 Time Dependent Perturbed Angular Distribution (TDPAD) Method

If a static magnetic field is placed perpendicular to the axial symmetry axis of the spin orientation, the Larmor precession of the isomeric spins in the applied field can be observed as a function of time, provided that the precession period is of the same order as the isomeric lifetime (or shorter). This method is called time-differential perturbed angular distribution (TDPAD).

The number of counts in a detector is calculated by using the formula

an ensemble of spin oriented radioactive nuclei with lifetime τ, in a direction (θ, φ) with respect to the LAB system. In this expression, Ak are the radiation parameters describing the type of radiation and its properties, Yk are the spherical harmonics which depend on the position of the detector (θ, φ) with respect to the LAB system and Bk are the orientation tensor describes the spin orientation of the ensemble with respect to the LAB system.

Fig. 4: The position of a detector in cylindrical co-ordinate system.

The external magnetic field used in TDPAD method depends on the isomeric lifetime and it varies over wide range.

Fig. 5: The Schematic drawing of the TDPAD experimental setup @ GANIL, France. The beam passes through a 50 m plastic scintillator before being stopped in a 500 m Cu foil.

R(t) function of each transition was generated by combining the data from detectors positioned at 90′ with respect to each other. It is given by

A graph between R(t) and T for two transitions is shown in figure 7. In this figure the pposite sign for 207 & 654 keV shows different multipolarity. 654 keV transition was found to be pure M2 and 207 keV was pure M1 transition.

Summary : In nuclei the electromagnetic moments are important nuclear properties. There are number of ways by which the magnetic moments and electric quadrupole moments in nuclei can be measured experimentally. The electro-magnetic moments in nuclei can be measured by various techniques based on the nature and the lifetime of the nucleus. To measure nuclear moments, the idea is to perturb the energy of the nucleus in the state of interest either by using internal (or external) magnetic field (in case of magnetic dipole moment) or by internal electric field (in case of electric quadrupole moment). The nuclear moments are then measured by analyzing the rotating angular distribution of de-excited gamma rays collected by suitably placed detectors.

References:

Introduction to Nuclear Physics – by Keneth S Krane.
Introductory Nuclear Physics – by Samuel S M Wong.
Nuclear Physics – by R R Roy & B P Nigam.
Handbook of Physics by Condon and Odishaw, TMH NewYork.
Introduction to Nuclear Physics, 2nd Edition, W.N.Cottingham & D.A. Greenwood.
Concept of Nuclear Physics by B L Cohen, McGraw Hill.
Nuclear Physics ; an Introduction by S.B. Patel.
Experimental techniques in Nuclear Physics by Dorin N. Poenaru & Walter Greiner
Exotic Nuclear Excitation by S.C. Pancholi
Nuclear spectroscopy Part B, by Fay Ajzenberg- Selove
Theory and Problems of modern Physics (Schaum’s outline Series)
Basic Ideas & Concepts in Nuclear Physics – by K Heyde
The “Particles of Modern Physics” by J. D. Stranathan, Philadephia: Blakiston.
Nuclear Physics by Irving Kaplan, Narosa Publishing House.

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Did you know ?

The nuclear moments are very important for understanding the structure of nuclei. The nuclear moments reveal information that is not or only indirectly available from other properties. They allow confirmation of hypotheses which were based on indirect experimental evidence, or they can be a very valuable input in nuclear models for determination and testing of the model parameters.
Nuclear magnetic moments are very sensitive to which orbits are occupied by the valence particles (or holes). Magnetic moments thus provide a good test for the purity of a particular configuration. They are most sensitive to the orbits in which the unpaired particles are moving but very little sensitive to the number of paired particles or holes (as long as they are paired to zero spin)
There are various techniques available to measure nuclear moments. The choice of a particular echnique is guided primarily by the following three things : the nucleus to be studied (stable/unstable), the way of producing the nucleus and the lifetime of the nuclear state to be probed.
To measure the magnetic moment of a nuclear state, there are some techniques which measure the g-factor and there are some more which measure the magnetic moment directly.
The nuclear moments have been studied since the very beginning of nuclear structure physics. The earliest measurements date back to the 1950s,with the nuclear magnetic resonance (NMR) technique. The measurement of quadrupole is more difficult and challenging than magnetic moment measurements.
The first quadrupole moment measurement have been reported in 1960s but the more systematic studies on quadrupole moments of stable nuclei started only in the late 1970s, using mainly two techniques: the hyperfine structure of muonic x-rays or the atomic beam magnetic resonance method.
To measure the magnetic moment (or g-factor) of a particular nuclear states, the nucleus is subjected to an external/internal magnetic field, so that it precess (Larmor precession) with significant frequency.
Due to rotation, the angular distribution of the decaying radiation (beta or gamma) is roatted which is then measured and magnetic moment is found.
For quadrupole moment measurement as precession purely due to electric field are required hence the excited nuclei are implanted into a non-magnetic material having electric field gradient.
The TDPAD technique of measurement is applied to measure the magnetic moment or quadrupole moment of isomeric states.

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