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We investigate the mixed-spin (spin-1/2 and spin-1) antiferromagnetic Ising model in two dimensions through a real-space renormalization group transformation. The model includes nearest and next-nearest neighbor interactions and is suitable to describe ferrimagnetic ordering. We present the finite-temperature phase diagram of the model. The presence of the second-neighbor bonds enlarge the ferrimagnetic phase. Our results do not indicate the existence of a tricritical point, contrary to previous effective-field calculations.

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Annual effective dose due to the consumption of 40K, 226Ra, 228Ra and 228Th was estimated from high-resolution gamma spectrometry, food consumption data for the inhabitants of Rio de Janeiro City and dose coefficients published by the ICRP 119. A total of 31 samples of cereals, grains, vegetables, flours, liquid and perishables were analyzed. 40K was measured in all samples, and bean sample presented highest specific concentration 489.36 ± 23.70 Bq kg-1. The highest specific concentration for 226Ra, 228Ra and 228Th was measured in pumpkin (7.82 ± 1.09 Bq kg-1), carrot (30.18 ± 1.99 Bq kg-1) and beet (2.43 ± 0.48 Bq kg-1), respectively. The highest contribution to annual effective dose came from beans (556.3 μSv), potato (12.5 μSv), carrot (10.3 μSv), banana (4.7 μSv) and beet (3.3 μSv). It was observed that updates of daily food consumption values was the main reason for an up to 10-fold difference between the annual effective dose found in the present study and literature data.

Rio de Janeiro city is the second most populous city in Brazil with almost 7 million inhabitants, the local market offers a great diversity of foods, nuts and seasonings, national and imported. Seasonings are used worldwide, some have a common use as salt and sugar, besides others used locally as cheiro verde, a mixture of chives (Allium schoenoprasum) and parsley (Petroselinum crispum) used in Rio de Janeiro, Brazil. According to the Onternational Food Safety Authorities Network (2011), plants used as food commonly have K-40, Th-232 and U-238 and their progenies. On a variety of concentrations, naturally occurring radionuclides are present in every part of the earth and in the tissue of all living beings. Natural radionuclides can be found almost everywhere; in soil, water and atmosphere, subjecting human beings to a daily exposure. The K-40 is responsible for about 60% of the total annual effective dose due to ingestion (United Nations Scientific Committee on the Effects of Atomic Radiation, 2008). Therefore, it is a radionuclide of relevance both in the environmental point of view and radiological protection (Zhu, 2001). Gamma-ray spectrometry using high-purity germanium detectors (HPGe) is a procedure widely used for determining the concentrations of natural and artificial radionuclides in environmental samples. As a nondestructive technique, this method possesses advantages in multi-element analysis, simplified sample preparation (i.e., does not require any chemical separation process), and applicability for precise quantitative determination of the radioactive content in a sample. The most accurate method to determine the activity concentration of radionuclides is to use an adequate standard source with similar geometrical dimensions, density, and chemical compositions to the sample under study (Díaz & Vargas, 2008). This study aims to determine activities concentrations for the radionuclides K-40, Ra-226, Ra-228, Ra-224 in nuts and seasoning samples obtained in Rio de Janeiro City, Brazil, using a hyper pure germanium detector (HPGe) and the LabSDCS software for the calculation of self-absorption correction. 2 Materials and methods 2.1 Sample analyses The nuts and seasoning samples analyzed were acquired in informal street commerce of Rio de Janeiro city, Brazil. The samples analyzed are listed in Table 1, along with their botanical names and the physical characteristic in which they were analyzed.