I live in a northern climate and won't be planting my garden for 4-6 weeks.
Will burning sticks in my garden negatively effect the soil?
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If it doesn't contain manufactured materials, plastics or anti-rot pressure treatment, burnt woody material when thoroughly consumed will leave behind the wood mineral content and charcoal.
You're burning off carbohydrates, cellulose and lignins which are complex molecules of mostly hydrogen, carbon and with any protein content nitrogen.
Of course all the gaseous material that is liberated is Carbon Dioxide, Water and various Nitrogen Oxides along with unburned particulates. Air quality might be an issue .
Charcoal is a non-issue, however; the ash left behind will be highly alkaline with potash being a major component. Potash is a mixture of various potassium salts, potassium being one of the big three, something you will be testing as to soil content.
As long as you spread the ash around and test for P levels and pH in your soil, you are merely adding a soil amendment. On Alkaline soils or soils with too much potassium due to over generous soil amendment (proper level for crop good, too much poisonous to growth), you will have to be careful as you will be doing a tremendous multi-year Ryegrass cover crop operation, with cut at full stand and removal of said growth to the dump till the potassium level is depleted to healthy levels. Or an acidification program if Potassium is not at issue.
Our garden area was used as the woodstove ash dump by previous owners. I will attest to finding sections that had bad vegetable growth and P testing readily showed the problem.
Also save some of that ash in a couple three pound coffee cans for your car's trunk. In that northern climate you can find it to be an amazing traction aide on ice slick surfaces.
Average pH 10.4
Average wood ash content
Trace Elements (dependent on soil content in your area, some Madrone wood in our area generates Paris Green clinker in the ash ( a copper arsenite compound - yipes!))
What do you hope to accomplish by burning the sticks? And will you be burning the sticks directly in the garden, or burning the sticks in a central location, then moving the ash to the garden?
If you are just hoping to get rid of the brush without carting it off site, burning might be an option. You could also try hugelkulture (essentially, burying it with a ton of nitrogen-rich waste), but in my experience, you really do need a ton of organic waste to bury with it or it does not help your soil.
Also, I would advise against burning the sticks directly in the garden. The heat of the fire will kill off the microbes in the soil, which really isn't what you want to do. You might consider creating your own biocar in a pit near your garden, then moving the charcoal into the garden. The science doesn't seem to support most of the positive claims about biochar, but plenty of people report good results using it, so it probably wouldn't hurt to try.