1. Trace Element in Behavior in Crystallization Isotope Geochemistry Lecture 26

2. Trace Elements During Crystallization As for melting, we can imagine two possibilities: equilibrium and fractional crystallization. Equilibrium crystallization occurs when the total liquid and total solid remain in equilibrium throughout the crystallization. If we define X as the fraction crystallized, then The limit of trace element enrichment or depletion occurs when X = 1, when C ℓ /C˚ = 1/D. Fractional crystallization, which assumes only instantaneous equilibrium between solid and liquid, is a more applicable model of crystallization. In this case, trace element concentrations in the melt are governed by: If D is very large, C ℓ /C o approaches 0 as X approaches 1, and it approaches ∞ as X approaches 1 if D is very small. For multiphase crystallization, we can replace it with the bulk distribution coefficient as we defined it earlier.

3. Fractional Crystallization Though fractional crystallization can, in principle, produce extreme trace element enrichment, this rarely occurs. A melt that has crystallized 90% or more would have major element chemistry very different and very different partition coefficients, and usually larger ones. This limits the enrichment of incompatible elements. However, highly compatible elements (elements with solid/liquid partition coefficients greater than 1, such as Ni) do have concentrations that approach 0 in fractionated melts.

4. In Situ Crystallization We think about how magmas might crystallize in a magma chamber and use these same basic equations to build more sophisticated models. In in situ crystallization, we imagine crystallization occurs in a crystal-rich mush on the margin of the chamber, where heat is being lost. Melt in the mush connects with the free magma in the chamber only by diffusion. This leads to a less enrichment of incompatible elements as crystallization proceeds. o ƒ A is rhe fraction of returning liquid and C f is the concentration in that liquid. f is the fraction of liquid remaining (and returning to magma chamber in the mush zone).

5. RTF Magma Chambers We know a little about how volcanoes work. In particular, they don’t completely empty during eruption. On long time scales (thousands of years), we can consider them to be continually refilled, tapped (erupted), and undergoing fractional crystallization. Thus they are open systems. Our equations thus far have been for closed systems. Open systems can lead to different styles of enrichment, and, ultimately in extreme cases, steady-state concentrations after many cycles.

6. Trace Element Ratios Incompatible elements ratios are less sensitive to fractional crystallization and partial melting than are absolute abundances, particularly if they are of similar incompatibility. For large extents of melting, the ratio of two incompatible elements in a magma will be similar to that ratio in the magma source. One approach is to plot the ratio of two incompatible elements against the abundance of the least compatible of the two. This kind of plot is sometimes referred to as a process identification plot because fractional crystallization and partial melting have very different slopes. Crystallization produce rather flat slopes on such a diagram. Partial melting produces a steeper slope and the slope produced by aggregates of fractional melts is similar to that of equilibrium partial melting. In situ crystallization can produce a range of slopes depending on the value of ƒ, but a value of 0.25 is probably most reasonable.

7. Summary Incompatible element abundances are strong functions of degree of melting (more so for fractional), compatible element abundances are only weakly dependent on extent of melting. o Two simplifications are important for partial melting. When D ≪ F, the enrichment is 1/F for batch melting. If D is large (i.e., D ≫ F), the depletion of the element in the melt is rather insensitive to F. In either case, when F approaches 0, the maximum enrichment or depletion is 1/D. Moderate amounts of fractional crystallization do not have dramatic effects on incompatible element concentrations. Concentrations of highly compatible elements dramatically decrease with extent of fractional crystallization. o The RTF model does have significantly greater effects on incompatible element concentrations than simpler models, however. This all works out nicely: compatible elements are good qualitative indicators of the extent of fractional crystallization and incompatible elements are good indicators of the degree of melting. Both geochemical and experimental evidence indicates that alkali basalts and are the products of lower degrees of melting than tholeiites such as MORB, which are generally produced by ~8 to 15% average extent of melting. o Highly undersaturated rocks such as nephelinites are probably produced by the smallest degrees of melting (1% or less). Incompatible element ratios are less sensitive to fractional crystallization and partial melting than are absolute abundances, particularly if they are of similar incompatibility. o For relatively large extents of melting, the ratio of two incompatible elements in a magma will be similar to that ratio in the magma source.

8. Radiogenic Isotope Geochemistry Introduction & Physics of the Nucleus

9. Radiogenic Isotope Geochemistry Radiogenic isotope geochemistry has its origins in the late 19 th century with the discovery of radioactive decay and in the early 20 th century with the discovery of the nuclear structure of the atom. Its initial success was demonstrating the antiquity of the Earth - much older than physicists had thought (geologists won this one) because: o It provided a source of energy to explain heat flowing out of the Earth and o It provide a direct means of dating rocks, and hence geologic events, and putting numbers on the relative time scale geologists had worked out in the 19 th century. Before we begin, we need to review a bit of nuclear physics, which is important not only in understanding radiogenic isotope geochemistry, but also the origin of the elements and their abundances.

10. Some terms Z: Proton number (= atomic number) N: Neutron number A: mass number (= N+Z) M: atomic mass (not an integer, like the above). Isotopes: o nuclei with the same Z but different N Isobars: o nuclei with the same A, but different Z, N Isotone: o nuclei with same N, but different Z o (not used much)

11. Nuclear Forces Not all possible combinations of protons and neutrons result in stable nuclei. Typically for stable nuclei, N ≈ Z. o Thus a significant portion of the nucleus consists of protons, which tend to repel each other. Another force must exist that is stronger than coulomb repulsion at short distances. It must be negligible at larger distances, otherwise all matter would collapse into a single nucleus. o This force, called the nuclear force, is a manifestation of one of the fundamental forces of nature, called the strong force. o If this force is assigned a strength of 1, then the strengths of other forces are: electromagnetic 10 -2 ; weak force 10 -5 ; gravity 10 -39.

12. Nuclear Binding Energy A general physical principle is that the lowest energy configuration is the most stable. We would thus expect that if 4 He is stable relative to 2 free neutrons and 2free protons, 4 He must be a lower energy state compared with the free particles. If this is the case, then from Einstein’s mass–energy equivalence: E = mc 2 we predict that the 4 He nucleus will have less mass than 2 free neutrons and 2 free protons. It does in fact have less mass. From the principle that the lowest energy configurations are the most stable and the mass–energy equivalence, we should be able to predict the relative stability of various nuclei from their masses alone. The mass decrement of an atom is δ = W - M o where W is the sum of the mass of the constituent particles and M is the actual mass. For 4 He for example, the δ = 0.030306 u. Converting this to energy yields 28.29 MeV. This energy is known as the binding energy. Dividing by A, the mass number, or number of nucleons, gives the binding energy per nucleon, E b : The mass decrement for 4 He is about 1%, (10 -2 ). The mass decrement associated with binding an electron to a nucleus is of the order of 10 -8, so bonds between nucleons are about 10 6 times stronger than bonds between electrons and nuclei.

13. Nuclear Stability: The Liquid Drop Model Why are some nuclei more stable than others? The answer has to do with the forces between nucleons and how nucleons are organized within the nucleus. The simplest model nucleus is Bohr’s liquid-drop model. It assumes all nucleons have equivalent states and treats the binding between nucleons as similar to the binding between molecules in a liquid drop, in which there are three effects: a ‘volume’ energy: energy needed to unbind or evaporate the nucleus: proportional to # of nucleons. a surface energy: a nucleon in the interior of the nucleus is surrounded by other nucleons and exerts no force on more distance nucleons. But at the surface, the force is unsaturated, leading to a force similar to surface tension. This force tends to minimize the surface area of the nucleus and is strongest for light nuclei and becomes rapidly less important for heavier nuclei. a coulomb energy: repulsive force between protons. It is proportional to the total number of proton pairs (Z(Z - 1)/2) and inversely proportional to radius

14. Shell Model ZNA (Z+N) Number of stable nuclei Number of very long- lived nuclei odd even45 oddevenodd503 evenodd 553 even 16511 Two observations suggest nucleons do not have equivalent states and that instead the nucleus has some kind of structure: o Nuclei with even number of protons and/or neutrons are more stable than those with odd numbers. o Nuclei with magic numbers of nucleons 2, 8, 20, 28, 50, 82, and 126) are more stable and stable nuclei with such magic numbers are particularly common. These observations lead to the shell model of the nucleus which is much like the shell model of the atom. o Two nucleons can be accepted into each ‘orbit’ and the nucleus is more stable when this happens and the spins cancel. o Nuclei, like atoms, are particularly stable when a ‘shell’ (consisting of a variety of orbits) is filled.

15. Decay of excited and unstable nuclei If we randomly throw protons and neutrons together to try to form a nucleus, there are three possible outcomes: o No nucleus can form from that particular combination of Z and N o A nucleus will form and not decay for periods much longer than the age of the universe o A nucleus forms, but will eventually decay by transmuting itself into a different nucleus with different N and Z. “Eventually” can be anything from 10 -12 sec to >10 15 years. Like atoms, nuclei can also exist in excited states, from which they ultimately decay. Decay of unstable nuclei (‘radioactive decay’) is much more energetic than decay of excited atoms and can involve either photons or particles or both (usually the latter). o Radioactive decay must obey all conservation laws (although it is mass-energy that is conserved, not just mass or energy).

16. Radioactive Decay There are 5 modes of radioactive decay Gamma o Emission of a high energy photon; usually accompanies other kinds of decay, but can occur when an excited nucleus decays to its ground state. Beta o Emission of a positron (β + ) or electron (β – ) as a proton transforms to a neutron or visa versa. In addition, a neutrino (ν) is also emitted. Its emission is necessary to conserve energy and spin (angular momentum). o Because of the nuclear structure and the way the elements were created, β - is more common that β +. o A remains unchanged but N and Z change - hence the element changes. Electron capture o An inner electron is captured, transforming a proton to a neutron - same effect as β + decay. X-rays emitted as outer electrons cascade inward. Alpha o Emission of a helium nucleus or α-particle. Reduces N and Z by 2 and A by 4. Only occurs in elements heavier than Fe (and in fact 147 Sm is the lightest naturally occurring nuclide that undergoes α-decay). Spontaneous Fission o Nucleus splits into subequal parts plus free neutrons. Only occurs in very heavy nuclei. Only occurs in 238 U among naturally occurring nuclei - and even then it is extremely rare. o Capture of one of these neutrons will induce fission in 235 U.

17. Beta Decay

18. Alpha & Gamma Decay

19. Basics of Radiogenic Isotope Geochemistry What makes radioactive decay useful to geochemists is that it occurs at a rate that is constant and completely independent of external influences*. The probability that a nucleus will decay is expressed by the decay constant, λ, which has units of inverse time and is unique to each radioactive nuclide. The rate of decay is given by the basic equation of radioactive decay: where N is the number of radioactive nuclides. This is a first order rate equation, like the ones we saw in kinetics. But unlike the rate constant, k, of kinetics, λ is a true constant and independent of everything. This is the only equation we need in radiogenic isotope geochemistry!. Because we can derived a whole bunch of other equations from it. 

20. Basics of Radiogenic Isotope Geochemistry We start with: Rearrange and integrate: and get:or o Half-life: Our radionuclide will decay to a radiogenic daughter so that D = N 0 - N and and usually there will have been some daughter around to begin with, D 0, so our equation is: D t N t