Briefly explain why the hardness of tempered martensite diminishes with tempering time (at constant temperature) and with increasing tem- perature (at constant tempering time).

Tempering is a critical process in metallurgy that helps in enhancing the ductility and toughness of martensite, a phase formed in steel during the rapid quenching process. Although martensite is known for its high hardness and strength, it's typically too brittle for most applications. By tempering, we heat the steel to temperatures below the eutectoid temperature, allowing for some of the trapped carbon atoms to diffuse out and the lattice structure of the iron to relax from body-centered tetragonal (BCT) to a more stable body-centered cubic (BCC) form. This relaxation results in a loss of some of the hardness that makes martensite advantageous for certain uses.

During tempering, the material undergoes transformations that balance strength and flexibility. Tempering provides a means of controlling the hardness and mechanical properties of steel by carefully selecting the temperature and duration of the heat treatment. Lower temperatures typically produce a harder steel, while higher temperatures lead to increased toughness but lower hardness.

Carbon plays a pivotal role in the hardness of martensite. Martensite's body-centered tetragonal structure is a supersaturated solid solution of carbon. Carbon diffusion is a key factor influencing the tempering of martensite. Upon heating, carbon atoms begin to move out of their interstitial sites within the iron lattice. This diffusion is time-dependent and gradually reduces the lattice strain and supersaturation, which, in turn, decreases hardness.

As tempering progresses, carbon atoms migrate and eventually form carbides. The mobility of carbon atoms increases with temperature, leading to faster diffusion at higher tempering temperatures. This is one of the primary reasons why increasing tempering temperature or time softens martensite. The diffusion of carbon is a transformative step that paves the way for carbide precipitation and ultimately reduces the hardness of the steel.

Carbide precipitation is an outcome of carbon diffusion during the tempering of martensite. As carbon atoms diffuse from their positions in the martensitic lattice, they combine with iron to form various types of carbides. The most common carbide to form is cementite (Fe₃C), although other carbides can also precipitate depending on the alloying elements present.

The precipitation of carbides redistributes the reinforced carbon from the martensitic matrix, reducing hardness and increasing toughness. The size, distribution, and type of carbides that form are crucial to the final properties of the steel. Initially, smaller carbides increase hardness by impeding dislocations in the lattice. Over time, these carbides can coalesce into larger particles, a process known as coarsening, which makes them less effective at blocking dislocations and therefore reduces hardness.

The tempering of martensite inevitably leads to numerous microstructural changes in steel. Apart from carbon diffusion and carbide precipitation, which have already been discussed, there are other transformations that occur within the metal. Upon tempering, martensite starts to convert into a mixture of ferrite and carbides, known as tempered martensite. This conversion process is gradual; increasing the tempering time or temperature hastens this evolution.

The initial hard martensitic structure becomes softer and more ductile, with a decrease in dislocation density. The formation of Fe₃C carbides and their growth play a significant part in the decrease of hardness over time. Higher tempering temperatures might also lead to the formation of retained austenite, another factor that can influence the overall hardness and toughness of the material. These microstructural changes dictate the mechanical properties of the steel and are extremely important in tailoring materials to their specific applications.